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P. Turán introduced [[#References|[a52]]] and developed (see [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a16]]], [[#References|[a21]]], [[#References|[a22]]], [[#References|[a23]]], [[#References|[a24]]], [[#References|[a25]]], [[#References|[a26]]], [[#References|[a27]]], [[#References|[a28]]], [[#References|[a29]]], [[#References|[a30]]], [[#References|[a31]]], [[#References|[a32]]], [[#References|[a33]]], [[#References|[a34]]], [[#References|[a35]]], [[#References|[a36]]], [[#References|[a37]]], [[#References|[a38]]], [[#References|[a39]]], [[#References|[a40]]], [[#References|[a41]]], [[#References|[a46]]], and all papers by Turán mentioned below) the power sum method, by which one can investigate certain minimax problems described below. The method is used in many problems of analytic [[Number theory|number theory]], analysis and applied mathematics.
 
P. Turán introduced [[#References|[a52]]] and developed (see [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a16]]], [[#References|[a21]]], [[#References|[a22]]], [[#References|[a23]]], [[#References|[a24]]], [[#References|[a25]]], [[#References|[a26]]], [[#References|[a27]]], [[#References|[a28]]], [[#References|[a29]]], [[#References|[a30]]], [[#References|[a31]]], [[#References|[a32]]], [[#References|[a33]]], [[#References|[a34]]], [[#References|[a35]]], [[#References|[a36]]], [[#References|[a37]]], [[#References|[a38]]], [[#References|[a39]]], [[#References|[a40]]], [[#References|[a41]]], [[#References|[a46]]], and all papers by Turán mentioned below) the power sum method, by which one can investigate certain minimax problems described below. The method is used in many problems of analytic [[Number theory|number theory]], analysis and applied mathematics.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t1202001.png" /> be a fixed set of integers. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t1202002.png" /> be fixed complex numbers and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t1202003.png" /> be complex numbers from a prescribed set. Define the following norms:
+
Let $S$ be a fixed set of integers. Let $b _ { j }$ be fixed complex numbers and let $z_j$ be complex numbers from a prescribed set. Define the following norms:
  
Bohr norm: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t1202004.png" />;
+
Bohr norm: $M _ { 0 } (  k  ) = \sum _ { j = 1 } ^ { n } | b _ { j } \| z _ { j } | ^ { k }$;
  
minimum norm: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t1202005.png" />;
+
minimum norm: $M _ { 1 } ( k ) = \operatorname { min } _ { j } | z _ { j } | ^ { k }$;
  
maximum norm: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t1202006.png" />;
+
maximum norm: $M _ { 2 } ( k ) = \operatorname { max } _ { j } | z _ { j } | ^ { k }$;
  
Wiener norm: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t1202007.png" />;
+
Wiener norm: $M _ { 3 } ( k ) = \left( \sum _ { j = 1 } ^ { n } | b _ { j } | ^ { 2 } | z _ { j } | ^ { 2 k } \right) ^ { 1 / 2 }$;
  
separation norm: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t1202008.png" />;
+
separation norm: $M _ { 4 } = \operatorname { min } _ { 1 \leq j &lt; k \leq n } | z _ { j } - z _ { k } |$;
  
Cauchy norm: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t1202009.png" />;
+
Cauchy norm: $M _ { 5 } = \operatorname { max } _ { j } | b _ { j } |$;
  
argument norm: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020010.png" />. Turán's method deals with the following problems [[#References|[a91]]].
+
argument norm: $M _ { 6 } = \operatorname { min } _ { j } | \operatorname { arc } z _ { j } |$. Turán's method deals with the following problems [[#References|[a91]]].
  
1) Determine, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020011.png" />,
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1) Determine, for $d \in [ 0,3 ]$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k \in S } \frac { | \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } | } { M _ { d } ( k ) }, \end{equation}
  
where the infimum is taken over all complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020013.png" /> (two-sided direct problems).
+
where the infimum is taken over all complex numbers $z_j$ (two-sided direct problems).
  
2) Find the above minimum in (a1) over all complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020014.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020015.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020016.png" /> ( "two-sided conditional problems" ).
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2) Find the above minimum in (a1) over all complex numbers $z_j$ satisfying $M _ { 4 } \geq \delta &gt; 0$ or $M _ { 6 } \geq \kappa &gt; 0$ ( "two-sided conditional problems" ).
  
3) For a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020018.png" />, find
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3) For a given domain $U$ and $d \in [ 0,3 ]$, find
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020019.png" /></td> </tr></table>
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\begin{equation*} \operatorname { inf } _ { z _ { j } \in U } \operatorname { max } _ { k \in S } \frac { \operatorname { Re } \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } } { M _ { d } ( k ) } \end{equation*}
  
 
(one-sided conditional problems).
 
(one-sided conditional problems).
  
4) For a given weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020021.png" />, find
+
4) For a given weight function $\psi ( k , n ) &gt; 0$ and $d \in [ 0,3 ]$, find
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020022.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k \in S } \left( \frac { | \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } | \psi ( k , n ) } { M _ { d } ( k ) } \right) ^ { 1 / k } \end{equation*}
  
 
(weighted two-sided problems).
 
(weighted two-sided problems).
  
5) For a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020024.png" />, find
+
5) For a given domain $U$ and $0 \leq d \leq 3$, find
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020025.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { sup } _ { z _ { 1 } , \ldots , z _ { n } \in U } \operatorname { min } _ { k \in S } \frac { | \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } | } { M _ { d } ( k ) } \end{equation*}
  
 
(dual conditional problems).
 
(dual conditional problems).
  
6) Given polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020030.png" />, determine
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6) Given polynomials $\phi ( x )$ and $\phi _ { j } ( x )$, $d \in [ 0,3 ]$, $g _ { 1 } ( k ) = \sum _ { j = 1 } ^ { n } \phi _ { j } ( k ) z _ { j } ^ { k }$ and $g _ 2 ( k ) = \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } \phi ( z _ { j } )$, determine
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020031.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k \in S } \frac { | g _ { 1 } ( k ) | } { M _ { d } ( k ) } \end{equation*}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020032.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k \in S } \frac { | g _ { 2 } ( k ) | } { M _ { d } ( k ) } \end{equation*}
  
 
(two-sided direct operator problems).
 
(two-sided direct operator problems).
  
7) Given a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020034.png" />, find
+
7) Given a domain $U$ and $d \in [ 0,3 ]$, find
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020035.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { inf } _ { z _ { 1 } , \ldots , z _ { n } \in U } \operatorname { max } _ { k \in S } \frac { \operatorname { Re } g _ { 1 } ( k ) } { M _ { d } (  k  ) } \end{equation*}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020036.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k \in S } \frac { \operatorname { Re } g _ { 2 } ( k ) } { M _ { d } ( k ) }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020038.png" /> are as above (one-sided conditional operator problems).
+
where $g _ { 1 } ( k )$ and $g_2 ( k )$ are as above (one-sided conditional operator problems).
  
8) Given a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020039.png" /> of integers, fixed complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020041.png" />, and two generalized power sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020043.png" />, how large can the quantities
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8) Given a finite set $S$ of integers, fixed complex numbers $b _ { j }$, $d \in [ 0,3 ]$, and two generalized power sums $g _ { 1 } ( k ) = \sum _ { j = 1 } ^ { n } b _ { j } ^ { \prime } ( k ) z _ { j } ^ { k }$, $g_2 ( k ) = \sum _ { j = 1 } ^ { n } b _ { j } ^ { \prime \prime } ( k ) z _ { j } ^ { k }$, how large can the quantities
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020044.png" /></td> </tr></table>
+
\begin{equation*} \frac { | g _ { 1 } ( k ) | } { M _ { d ^ { \prime } } ( k ) } , \frac { | g _ { 2 } ( k ) | } { M _ { d ^ { \prime \prime } } ( k ) } \quad ( k \in S ) \end{equation*}
  
be made simultaneously depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020048.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020049.png" /> (simultaneous problems)?
+
be made simultaneously depending only on $b _ { j }$, $d ^ { \prime }$, $d ^ { \prime \prime }$, $n$, and $S$ (simultaneous problems)?
  
9) Given two finite sets of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020051.png" />, fixed complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020055.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020056.png" />, what is
+
9) Given two finite sets of integers $S _ { 1 }$ and $S _ { 2 }$, fixed complex numbers $b _ { j }$, $h ( m , k ) = \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } w _ { j }^ { m }$, $| z _ { 1 } | \geq \ldots \geq | z _ { n } |$, $| w _ { 1 } | \geq \ldots \geq | w _ { n } |$, and $0 \leq d ^ { \prime } , d ^ { \prime \prime } \leq 3$, what is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020057.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { inf } _ { z _ { j } , w _ { j } } \operatorname { max } _ { k \in S _ { 1 } , \atop m \in S _ { 2 } } \frac { | h ( m , k ) | } { M _ { d ^ { \prime } }  ( k ) M _ { d^ { \prime \prime } }  ( m ) } \end{equation*}
  
 
and what are the extremal systems (several variables problems)?
 
and what are the extremal systems (several variables problems)?
Line 77: Line 85:
 
Turán and others obtained some lower bounds for some of the above problems.
 
Turán and others obtained some lower bounds for some of the above problems.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020058.png" /> be a pure power sum. Then
+
Let $s _ { k } = z _ { 1 } ^ { k } + \ldots + z _ { n } ^ { k }$ be a pure power sum. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020059.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k = 1 , \ldots , n } \frac { | s _ { k } | } { M _ { 1 } ( k ) } = 1 \end{equation*}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020060.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k = 1 , \ldots , 2 n - 1 } \frac { | s _ { k } | } { M _ { 2 } ( k ) } = 1 \end{equation*}
  
(see also [[#References|[a4]]]). These results were obtained in the equivalent form with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020062.png" />, respectively.
+
(see also [[#References|[a4]]]). These results were obtained in the equivalent form with $M _ { 1 } ( k ) = 1$ and $M _ { 2 } ( k ) = 1$, respectively.
  
Also, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020064.png" />. Then
+
Also, let $R _ { n } = \operatorname { min } _ { z _ { j } } \operatorname { max } _ { k = 1 , \ldots , n } | s _ { k } |$, where $\max| z _ { j } | = 1$. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020065.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} R _ { n } &gt; \frac { \operatorname { log } 2 } { 1 + \frac { 1 } { 2 } + \ldots + \frac { 1 } { n } }. \end{equation}
  
F.V. Atkinson [[#References|[a2]]] improved this by showing that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020066.png" />. A. Biro [[#References|[a3]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020067.png" /> and that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020068.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020070.png" />, then
+
F.V. Atkinson [[#References|[a2]]] improved this by showing that $R _ { n } &gt; 1 / 5$. A. Biro [[#References|[a3]]] proved that $R _ { n } &gt; 1 / 2$ and that if $m &gt; 0$ is such that $z _ { 1 } = \ldots = z _ { m } = 1$, $n \geq n _ { 0 }$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020071.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { max } _ { j = 1 , \ldots , n - m + 1 } | s _ { j } | \geq m \left( \frac { 1 } { 2 } + \frac { m } { 8 n } + \frac { 3 m ^ { 2 } } { 64 n ^ { 2 } } \right). \end{equation*}
  
J. Anderson [[#References|[a1]]] showed that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020072.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020073.png" />, and that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020074.png" /> is a [[Prime number|prime number]], then this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020075.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020076.png" />; he also proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020077.png" />, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020078.png" /> such that
+
J. Anderson [[#References|[a1]]] showed that if $\operatorname{min}_{j} | z _ { j } | = 1$, then $\operatorname { inf } _ { z _ { j } } \operatorname { max } _ { j = 1 , \ldots , n^2 }  | s _ { j } | \geq \sqrt { n }$, and that if $n + 1$ is a [[Prime number|prime number]], then this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020075.png"/> lies in $[ \sqrt { n } , \sqrt { n + 1 } ]$; he also proved that if $m \in [ 1 , n - 1 ]$, then there exists a $c = c ( m )$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020079.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { max } _ { r = 1 , \ldots , c n } \frac { | z _ { 1 } ^ { r } + \ldots + z _ { n } ^ { r } | } { \operatorname { min } _ { k = 1 , \ldots , n } | z _ { k } ^ { r } | } \geq m. \end{equation*}
  
It is also known [[#References|[a43]]] that, on the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020080.png" /> for infinitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020081.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020082.png" /> for large enough <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020083.png" />.
+
It is also known [[#References|[a43]]] that, on the other hand, $R _ { n } &lt; 1 - \operatorname { log } n / ( 3 n )$ for infinitely many $n$ and that $R _ { n } &lt; 1 - 1 / ( 250 n )$ for large enough $n$.
  
 
P. Erdös proved that
 
P. Erdös proved that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020084.png" /></td> </tr></table>
+
\begin{equation*} M _ { 2 } = \operatorname { min } _ { z _ { j } } \operatorname { max } _ { k = 2 , \ldots , n + 1 } | s _ { k } | \leq 2 ( n + 1 ) ^ { 2 } e ^ { - \theta n }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020085.png" /> is the solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020086.png" />, and L. Erdös [[#References|[a15]]] proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020087.png" /> is large enough, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020088.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020089.png" /> is the solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020090.png" />.
+
where $\theta \approx 0.2784$ is the solution of the equation $x \operatorname { exp } ( x + 1 ) = 1$, and L. Erdös [[#References|[a15]]] proved that if $n$ is large enough, then $\operatorname { exp } ( - 2 \theta n - 0.7823 \operatorname { log } n ) \leq M _ { 2 } \leq \operatorname { exp } ( - 2 \theta n + 4.5 \operatorname { log } n )$, where $\theta$ is the solution of the equation $1 + \theta + \operatorname { log } \theta = 0$.
  
 
E. Makai [[#References|[a44]]] showed that
 
E. Makai [[#References|[a44]]] showed that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020091.png" /></td> </tr></table>
+
\begin{equation*} M _ { 3 } = \operatorname { min } _ { z _ { j } } \operatorname { max } _ { k = 3 , \ldots , n + 2 } | s _ { k } | &lt; \frac { 1 } { 1.473 ^ { n } } \text { for } n &gt; n _ { 0 }. \end{equation*}
  
For generalized power sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020092.png" />, Turán proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020093.png" />, then
+
For generalized power sums $g ( k ) = \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k }$, Turán proved that if $\min_{ z _ { j }} | z _ { j } | = 1$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020094.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { max } _ { k = m + 1 , \ldots , m + n } | g ( k ) | \geq \left( \frac { n } { 2 e ( m + n ) } \right) ^ { n } | b _ { 1 } + \ldots + b _ { n } |. \end{equation*}
  
Makai [[#References|[a45]]] and N.G. de Bruijn [[#References|[a4]]] proved, independently, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020095.png" /> can be replaced with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020096.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020097.png" />. If, however, one replaces it with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020098.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t12020099.png" />, then the above inequality fails. Turán also proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200100.png" />, then
+
Makai [[#References|[a45]]] and N.G. de Bruijn [[#References|[a4]]] proved, independently, that $( n / ( 2 e ( m + n ) ) ) ^ { n }$ can be replaced with $1 / P _ { m , n }$, where $P _ { m , n } = \sum _ { j = 0 } ^ { n - 1 } \left( \begin{array} { c } { m + j } \\ { j } \end{array} \right) 2 ^ { j }$. If, however, one replaces it with $1 / ( P _ { m ,\, n } - \epsilon )$ for any $\epsilon &gt; 0$, then the above inequality fails. Turán also proved that if $\operatorname{min}_{j} | z _ { j } | = 1$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200101.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { max } _ { k = m + 1 , \ldots , m + n } | g ( k ) | \geq \frac { 1 } { 3 } | g ( 0 ) | \prod _ { j = 1 } ^ { n } \frac { | z _ { j } | - \operatorname { exp } ( - 1 / m ) } { | z _ { j } | + 1 }. \end{equation*}
  
G. Halasz showed that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200102.png" />,
+
G. Halasz showed that for any $k &gt; 1$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200103.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200103.png"/></td> </tr></table>
  
S. Gonek [[#References|[a18]]] proved that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200104.png" />,
+
S. Gonek [[#References|[a18]]] proved that for all $r &gt; 0$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200105.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { max } _ { 1 \leq k \leq 4  \left( \begin{array} { c } { n + r - 1 } \\ { r } \end{array} \right)} | g ( k ) | \geq | g ( 0 ) | \left( 2 e \left( \begin{array} { c } { n + r - 1 } \\ { r } \end{array} \right) \right) ^ { - 1 / r }. \end{equation*}
  
In the case of the maximum norm, V. Sos and Turán [[#References|[a46]]] obtained the following result. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200106.png" />. Then for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200107.png" />,
+
In the case of the maximum norm, V. Sos and Turán [[#References|[a46]]] obtained the following result. Let $1 = | z _ { 1 } | \geq \ldots \geq | z _ { n } |$. Then for any integer $m \geq 0$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200108.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { max } _ { k = m + 1 , \ldots , m + n } | g ( k ) | \geq c _ { m , n } , \operatorname { min } _ { j = 1 , \ldots , n } | b _ { 1 } + \ldots + b _ { j } | \end{equation*}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200109.png" />. G. Kolesnik and E.G. Straus [[#References|[a42]]] improved this by showing that one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200110.png" />. On the other hand, Makai [[#References|[a45]]] showed that for
+
with $c _ { m , n } = 2 ( n / ( 8 e ( m + n ) ) ) ^ { n }$. G. Kolesnik and E.G. Straus [[#References|[a42]]] improved this by showing that one can take $c _ { m , n } = \sqrt { n } ( n / ( 4 e ( m + n ) ) ) ^ { n }$. On the other hand, Makai [[#References|[a45]]] showed that for
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200111.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200111.png"/></td> </tr></table>
  
the inequality fails for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200113.png" />.
+
the inequality fails for some $m$ and $z_j$.
  
Considering different ranges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200114.png" />, Halasz [[#References|[a19]]] proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200115.png" />, then
+
Considering different ranges for $k$, Halasz [[#References|[a19]]] proved that if $m , n &lt; N$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200116.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { min } _ { k = m + 1 , \ldots , m + N } | g ( k ) | \geq \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200117.png" /></td> </tr></table>
+
\begin{equation*} \geq \frac { n } { 4 N ^ { 3  / 2} } \operatorname { exp } \left( - 30 n \left( \frac { 1 } { \operatorname { log } ( N / n ) } + \frac { 1 } { \operatorname { log } ( N / m ) } \right) \right) \times \times \operatorname { min } _ { l \leq n } \left| \sum _ { j = 1 } ^ { l } b _ { j }\right| . \end{equation*}
  
 
==Other norms and conditions.==
 
==Other norms and conditions.==
 
The following results are obtained for two-sided problems with other norms and conditions.
 
The following results are obtained for two-sided problems with other norms and conditions.
  
A) ([[#References|[a17]]], [[#References|[a47]]], [[#References|[a8]]], [[#References|[a45]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200118.png" /> be ordered so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200119.png" />. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200121.png" />. Then
+
A) ([[#References|[a17]]], [[#References|[a47]]], [[#References|[a8]]], [[#References|[a45]]]). Let $z_j$ be ordered so that $0 = | z _ { 1 } - 1 | \leq \ldots \leq | z _ { n } - 1 |$. Assume that $m \geq - 1$ and $n &gt; 1$. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200122.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { max } _ { k = m + 1 , \ldots , m + n } | g ( k ) | \geq \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200123.png" /></td> </tr></table>
+
\begin{equation*} \geq \frac { 1 } { 8 } \left( \frac { n - 1 } { 8 e ( m + n ) } \right) ^ { n } \operatorname { min }_ j | b _ { 1 } + \ldots + b _ { j } |. \end{equation*}
  
B) ([[#References|[a91]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200124.png" /> be ordered as in A). Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200126.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200127.png" /> be the largest integer satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200128.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200129.png" /> be the smallest integer satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200130.png" /> (if such an integer does not exist, take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200131.png" />). Then
+
B) ([[#References|[a91]]]). Let $z_j$ be ordered as in A). Assume that $m &gt; - 1$ and $0 &lt; \delta _ { 1 } &lt; \delta _ { 2 } &lt; n / ( m + n + 1 )$, let $h$ be the largest integer satisfying $| 1 - z _ { h } | &lt; \delta _ { 1 }$ and let $l$ be the smallest integer satisfying $| 1 - z _{l + 1} | &gt; \delta _ { 2 }$ (if such an integer does not exist, take $l = n$). Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200132.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { max } _ { k = m + 1 , \ldots , m + n } | g ( k ) | \geq \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200133.png" /></td> </tr></table>
+
\begin{equation*} \geq 2 \left( \frac { \delta _ { 1 } - \delta _ { 2 } } { 12 e } \right) ^ { n } \operatorname { min } _ { j = h , \ldots , l } | b _ { 1 } + \ldots + b _ { j } |. \end{equation*}
  
C) ([[#References|[a12]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200134.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200136.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200137.png" /> be such that
+
C) ([[#References|[a12]]]). Let $m \geq 0$ and let $k$, $k_{1}$, $k_2$ be such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200138.png" /></td> </tr></table>
+
\begin{equation*} | z _ { 1 } | \geq \ldots \geq | z _ { k _ { 1 } } | &gt; \frac { m + 2 n } { m + n } \geq \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200139.png" /></td> </tr></table>
+
\begin{equation*} \geq | z _ { k_1 } + 1 | \geq \ldots \geq | z _ { k } | = 1 \geq \ldots \geq | z _ { k _ { 2 } } - 1 | &gt; &gt; \frac { m } { m + n } \geq | z _ { k _ { 2 } } | \geq \ldots \geq | z _ { n } |. \end{equation*}
  
 
Then
 
Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200140.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { max } _ { r = m + 1 , \ldots , m + n } | g ( r ) | \geq \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200141.png" /></td> </tr></table>
+
\begin{equation*} \geq \frac { 1 } { n } \left( \frac { n } { 16 e ( m + n ) } \right) ^ { n } \times \times \operatorname{min} _ { k _ { 1 } \leq l _ { 1 } \leq k \leq l _ { 2 } \leq k _ { 2 } } | b _ {l_{ 1} } + \ldots + b _ {l_{  2 }} |. \end{equation*}
  
D) ([[#References|[a59]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200143.png" />, then
+
D) ([[#References|[a59]]]). If $m &gt; - 1$ and $\operatorname {min}_{ \mu \neq \nu} | z _ { \mu } - z _ { \nu } | \geq \delta \operatorname { max } _ { j } | z _ { j }|$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200144.png" /></td> </tr></table>
+
\begin{equation*} \frac { \max_{k = m + 1 , \ldots , m + n}| g ( k ) | } { \sum _ { j = 1 } ^ { n } | b _ { j } z _ { j } ^ { k } | } \geq \frac { 1 } { n } ( \frac { \delta } { 2 } ) ^ { n - 1 }. \end{equation*}
  
E) ([[#References|[a8]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200145.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200146.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200147.png" />, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200148.png" /> such that
+
E) ([[#References|[a8]]]). If $m &gt; - 1$ and $r$ is such that $\operatorname{min}_{j \neq r} | z _ { j } - z _ { r } | \geq \delta | z _ { r } |$, then there exists a $ k  \in [ m + 1 , m + n ]$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200149.png" /></td> </tr></table>
+
\begin{equation*} | g ( k ) | \geq ( \frac { \delta } { 2 + 2 \delta } ) ^ { n - 1 } | b _ { r } z _ { r} ^ { k } |. \end{equation*}
  
F) ((Halasz). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200150.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200151.png" /> be non-negative integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200152.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200153.png" />. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200154.png" />. Then there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200155.png" /> such that
+
F) ((Halasz). Let $m_1$ and $m _ { 2 }$ be non-negative integers, $m = \operatorname { max } ( m _ { 1 } , m _ { 2 } )$, and $S = [ - m _ { 1 } - n , - m _ { 1 } - 1 ] \cup [ m _ { 2 } + 1 , m _ { 2 } + n ]$. Assume that $z _ { 1 } \dots z _ { n } \neq 0$. Then there exists an integer $k \in S$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200156.png" /></td> </tr></table>
+
\begin{equation*} | g ( k ) | \geq \left( \frac { n } { 8 e ( m + n ) } \right) ^ { n } | g ( 0 ) |. \end{equation*}
  
G) (Turán). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200157.png" />, then the above inequality holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200158.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200159.png" />.
+
G) (Turán). If $S = [ m + 1 , m + n ] \cup [ 2 m + 1,2 m + n ]$, then the above inequality holds with $6$ instead of $8$.
  
 
==Problems of type 3) and 7).==
 
==Problems of type 3) and 7).==
Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200160.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200161.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200162.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200163.png" /> be real numbers, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200164.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200165.png" />. Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200166.png" /> for some fixed complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200167.png" />. Assuming that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200168.png" />, Turán proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200170.png" />, where
+
Assume that $\kappa \leq | \operatorname { arc } z _ { j } | \leq \pi$, $j = 1 , \ldots , n$, with $0 &lt; \kappa \leq \pi / 2$, let $a _ { j }$ be real numbers, and let $\phi ( z ) = z ^ { k } + a _ { 1 } z ^ { k - 1 } + \ldots + a _ { k } \neq 0$ for $| z | &gt; \rho \in ( 0,1 )$. Define $G _ { 2 } ( k ) = \sum _ { j = 1 } ^ { n } b _ { j } \phi ( z _ { j } ) z _ { j } ^ { k }$ for some fixed complex numbers $b _ { j }$. Assuming that $\operatorname{min}_{j} | z _ { j } | = 1$, Turán proved that $\max _ r \operatorname { Re } G _ { 2 } ( r ) \geq A$ and $\operatorname { min}_r \operatorname { Re } G _ { 2 } ( r ) \leq - A$, where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200171.png" /></td> </tr></table>
+
\begin{equation*} A = \frac { 1 } { 6 n 16 ^ { n } } \left( \frac { 1 + \rho } { 2 } \right) ^ { m } \left( \frac { 1 - \rho } { 2 } \right) ^ { 2 n + k } \left| \operatorname { Re } \sum _ { j = 1 } ^ { n } b _ { j } \right|  \end{equation*}
  
and the minimum is taken over all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200172.png" />.
+
and the minimum is taken over all integers $r \in [ m + 1 , m + n ( 3 + \pi / k ) ]$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200173.png" />, then the above inequalities hold with
+
If $\phi ( z ) = 1$, then the above inequalities hold with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200174.png" /></td> </tr></table>
+
\begin{equation*} A = \frac { 1 } { 6 n } \operatorname { min } _ { n \leq x \leq 2 n } \left( \frac { x } { 4 e ( m + x ) } \right) ^ { x } \left| \operatorname { Re } \sum _ { j = 1 } ^ { n } b _ { j } \right|. \end{equation*}
  
Also, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200175.png" /> are polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200176.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200177.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200178.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200179.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200180.png" />, where
+
Also, if $P _ { j } ( x )$ are polynomials of degree $k_j - 1$, $G _ { 1 } ( r ) = \sum _ { j = 1 } ^ { n } P _ { j } ( r ) z _ { j } ^ { r }$ and $K = k _ { 1 } + \ldots + k _ { n }$, then $\operatorname {max}_{r}\operatorname { Re } G _ { 1 } ( r ) \geq B$ and $\min_r \operatorname{Re} G _ { 1 } ( r ) \leq - B$, where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200181.png" /></td> </tr></table>
+
\begin{equation*} B = \frac { 1 } { 6 K } \left( \frac { K } { 4 e ( m + 2 K ) } \right) ^ { 2 K } \left| \operatorname { Re } \sum _ { j = 0 } ^ { n } P _ { j } ( 0 ) \right| \end{equation*}
  
and the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200182.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200183.png" />.
+
and the range of $r$ is $[ m + 1 , m + K ( 3 + \pi / \kappa ) ]$.
  
Assume now that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200184.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200185.png" /> be as defined above, and assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200186.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200187.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200188.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200189.png" />. Assume also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200190.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200191.png" />. Take any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200192.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200193.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200194.png" /> and define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200195.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200196.png" /> by
+
Assume now that $\operatorname {max}_{j} | z _ { j } | = 1$. Let $G _ { 2 } ( r )$ be as defined above, and assume $\phi ( z ) \neq 0$ for $z \in \{ | z | \geq \rho \} \cup \{ | \operatorname { arc } z | &lt; \kappa \}$, where $0 &lt; \rho &lt; 1$ and $0 &lt; \kappa &lt; \pi / 2$. Assume also that $\kappa \leq | \operatorname { arc } z _ { j } | &lt; \pi$, $j = 1 , \ldots , n$. Take any $\delta _ { 1 }$, $\delta _ { 2 }$ satisfying $1 &gt; \delta _ { 1 } &gt; \delta _ { 2 } \geq \rho$ and define $h_{1}$, $h _ { 2 }$ by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200197.png" /></td> </tr></table>
+
\begin{equation*} 1 = | z _ { 1 } | \geq \ldots \geq | z _ { h _ { 1 } } | \geq \delta _ { 1 } &gt; \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200198.png" /></td> </tr></table>
+
\begin{equation*} &gt; | z _ { h _ { 1 } } + 1 | \geq \ldots \geq | z _ { h _ { 2 } } | &gt; \delta _ { 2 } \geq \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200199.png" /></td> </tr></table>
+
\begin{equation*} \geq | z _ { h_2 } + 1 | \geq \ldots \geq | z _ { n } |. \end{equation*}
  
(If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200200.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200201.png" /> do not exist, replace them with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200202.png" />.) Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200203.png" /> and
+
(If $h_{1}$ or $h _ { 2 }$ do not exist, replace them with $n$.) Put $I = [ m + 1 , m + ( n + k ) ( 3 + \pi / k ) ]$ and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200204.png" /></td> </tr></table>
+
\begin{equation*} M = \frac { 1 } { 3 ( n + k ) } \left( \frac { \delta _ { 1 } - \delta _ { 2 } } { 16 } \right) ^ { 2 n + 2 k } \delta _ { 2 } ^ { m + ( n + k ) ( 1 + \pi / k ) }\times \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200205.png" /></td> </tr></table>
+
\begin{equation*} \times \operatorname { min } _ { h _ { 1 } \leq j \leq h _ { 2 } } | \operatorname { Re } ( b _ { 1 } + \ldots + b _ { j } ) |. \end{equation*}
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200207.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200208.png" />, then the above result holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200209.png" />.
+
Then $\max_{r \in I} \text{Re} \, G_2 (r ) \geq M$ and $\operatorname{min}_{r\in I} \operatorname{Re} G _ { 2 } ( r ) \leq - M$. If $\phi ( z ) = 1$, then the above result holds with $k = \rho = 0$.
  
J.D. Buchholtz [[#References|[a5]]], [[#References|[a6]]] proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200210.png" />, then
+
J.D. Buchholtz [[#References|[a5]]], [[#References|[a6]]] proved that if $\operatorname {max}_{j} | z _ { j } | = 1$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200211.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { max } _ { k = 1 , \ldots , n } \left( \frac { 1 } { n } | s _ { k } | \right) ^ { 1 / k } &gt; \frac { 1 } { 5 } &gt; \frac { 1 } { 2 + \sqrt { 8 } }, \end{equation*}
  
 
respectively, where the last result is the best possible.
 
respectively, where the last result is the best possible.
Line 225: Line 233:
 
R. Tijdeman [[#References|[a47]]] proved the following result for  "operator-type problems" .
 
R. Tijdeman [[#References|[a47]]] proved the following result for  "operator-type problems" .
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200212.png" /> be fixed complex polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200213.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200214.png" />. Then for every integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200215.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200216.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200217.png" />, the inequality
+
Let $P _ { j } ( x )$ be fixed complex polynomials of degree $k_j - 1$ and let $G _ { 1 } ( k ) = \sum _ { j = 1 } ^ { n } P _ { j } ( k ) z _ { j } ^ { k }$. Then for every integer $m \geq 0$, $K = k _ { 1 } + \ldots + k _ { n }$, and $\operatorname{min}_{j} | z _ { j } | = 1$, the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200218.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} \operatorname { min } _ { r = m + 1 , \ldots , m + K } | G _ { 1 } ( r ) | \geq \frac { 1 } { P _ { m , K } } \left| \sum _ { j = 1 } ^ { n } P _ { j } ( 0 ) \right| \end{equation}
  
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200219.png" /> is defined above and the factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200220.png" /> is the best possible; also, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200221.png" />, then (a3) holds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200222.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200223.png" />.
+
holds, where $P _ { m , K }$ is defined above and the factor $1 / P _ { m  , K}$ is the best possible; also, if $\operatorname {max}_{j} | z _ { j } | = 1$, then (a3) holds with $( K / ( 8 e ( m + K ) ) ) ^ { K }$ instead of $1 / P _ { m  , K}$.
  
 
J. Geysel [[#References|[a17]]] improved the above constant to
 
J. Geysel [[#References|[a17]]] improved the above constant to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200224.png" /></td> </tr></table>
+
\begin{equation*} \frac { 1 } { 4 } \left( \frac { K - 1 } { 8 e ( m + K ) } \right) ^ { K }. \end{equation*}
  
Turán studied the other  "operator-type problem"  for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200225.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200226.png" /> be fixed complex numbers and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200227.png" /> be a polynomial with no zeros outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200228.png" />. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200229.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200230.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200231.png" />. Then
+
Turán studied the other  "operator-type problem"  for $G _ { 2 } ( r ) = \sum _ { j = 1 } ^ { n } b _ { j } \phi ( z _ { j } ) z _ { j } ^ { k }$. Let $b _ { j }$ be fixed complex numbers and let $\phi ( z ) = z ^ { k } + a _ { 1 } z ^ { k - 1 } + \ldots + a _ { k }$ be a polynomial with no zeros outside $| z | &lt; \rho$. Assume that $m &gt; - 1$, $0 &lt; \rho &lt; 1$ and $\operatorname {max}_{j} | z _ { j } | = 1$. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200232.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200232.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a4)</td></tr></table>
  
 
with
 
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200233.png" /></td> </tr></table>
+
\begin{equation*} c _ { m , n } = 2 ^ { - n } \left( \frac { 1 + \rho } { 2 } \right) ^ { m } \left( \frac { 1 - \rho } { 2 } \right) ^ { n + k }. \end{equation*}
  
In case of the maximum norm and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200234.png" />, Turán proved (a4) with
+
In case of the maximum norm and $1 = | z _ { 1 } | \geq \ldots \geq | z _ { n } | &gt; 0$, Turán proved (a4) with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200235.png" /></td> </tr></table>
+
\begin{equation*} c _ { m ,\, n } = \left\{ \begin{array} { l l } { 2 ^ { 1 - n } \left( \frac { n + k } { 4 e ( m + n + k ) } \right) ^ { n + k } } &amp; { \text { if } \frac { m } { m + n + k } \geq \rho, } \\ { \rho ^ { m } 2 ^ { 1 - n } \left( \frac { 1 - \rho } { 4 } \right) ^ { n + k } } &amp; { \text { if } \frac { m } { m + n + k } &lt; \rho. } \end{array} \right. \end{equation*}
  
He also proved the following  "simultaneous problem" . Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200236.png" />. For any integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200237.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200238.png" /> there exist a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200239.png" /> such that the inequalities
+
He also proved the following  "simultaneous problem" . Let $\operatorname{min}_j | z _ { j } | = \operatorname { min } _ { j } | w _ { j } | = 1$. For any integers $m &gt; - 1$ and $n _ { 1 } , n _ { 2 } \geq 1$ there exist a $ k  \in [ m + 1 , m + n _ { 1 } n _ { 2 } ]$ such that the inequalities
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200240.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200240.png"/></td> </tr></table>
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200241.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120200/t120200241.png"/></td> </tr></table>
  
 
hold simultaneously.
 
hold simultaneously.
  
 
====References====
 
====References====
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+
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Revision as of 16:46, 1 July 2020

P. Turán introduced [a52] and developed (see [a9], [a10], [a11], [a12], [a13], [a14], [a16], [a21], [a22], [a23], [a24], [a25], [a26], [a27], [a28], [a29], [a30], [a31], [a32], [a33], [a34], [a35], [a36], [a37], [a38], [a39], [a40], [a41], [a46], and all papers by Turán mentioned below) the power sum method, by which one can investigate certain minimax problems described below. The method is used in many problems of analytic number theory, analysis and applied mathematics.

Let $S$ be a fixed set of integers. Let $b _ { j }$ be fixed complex numbers and let $z_j$ be complex numbers from a prescribed set. Define the following norms:

Bohr norm: $M _ { 0 } ( k ) = \sum _ { j = 1 } ^ { n } | b _ { j } \| z _ { j } | ^ { k }$;

minimum norm: $M _ { 1 } ( k ) = \operatorname { min } _ { j } | z _ { j } | ^ { k }$;

maximum norm: $M _ { 2 } ( k ) = \operatorname { max } _ { j } | z _ { j } | ^ { k }$;

Wiener norm: $M _ { 3 } ( k ) = \left( \sum _ { j = 1 } ^ { n } | b _ { j } | ^ { 2 } | z _ { j } | ^ { 2 k } \right) ^ { 1 / 2 }$;

separation norm: $M _ { 4 } = \operatorname { min } _ { 1 \leq j < k \leq n } | z _ { j } - z _ { k } |$;

Cauchy norm: $M _ { 5 } = \operatorname { max } _ { j } | b _ { j } |$;

argument norm: $M _ { 6 } = \operatorname { min } _ { j } | \operatorname { arc } z _ { j } |$. Turán's method deals with the following problems [a91].

1) Determine, for $d \in [ 0,3 ]$,

\begin{equation} \tag{a1} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k \in S } \frac { | \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } | } { M _ { d } ( k ) }, \end{equation}

where the infimum is taken over all complex numbers $z_j$ (two-sided direct problems).

2) Find the above minimum in (a1) over all complex numbers $z_j$ satisfying $M _ { 4 } \geq \delta > 0$ or $M _ { 6 } \geq \kappa > 0$ ( "two-sided conditional problems" ).

3) For a given domain $U$ and $d \in [ 0,3 ]$, find

\begin{equation*} \operatorname { inf } _ { z _ { j } \in U } \operatorname { max } _ { k \in S } \frac { \operatorname { Re } \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } } { M _ { d } ( k ) } \end{equation*}

(one-sided conditional problems).

4) For a given weight function $\psi ( k , n ) > 0$ and $d \in [ 0,3 ]$, find

\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k \in S } \left( \frac { | \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } | \psi ( k , n ) } { M _ { d } ( k ) } \right) ^ { 1 / k } \end{equation*}

(weighted two-sided problems).

5) For a given domain $U$ and $0 \leq d \leq 3$, find

\begin{equation*} \operatorname { sup } _ { z _ { 1 } , \ldots , z _ { n } \in U } \operatorname { min } _ { k \in S } \frac { | \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } | } { M _ { d } ( k ) } \end{equation*}

(dual conditional problems).

6) Given polynomials $\phi ( x )$ and $\phi _ { j } ( x )$, $d \in [ 0,3 ]$, $g _ { 1 } ( k ) = \sum _ { j = 1 } ^ { n } \phi _ { j } ( k ) z _ { j } ^ { k }$ and $g _ 2 ( k ) = \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } \phi ( z _ { j } )$, determine

\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k \in S } \frac { | g _ { 1 } ( k ) | } { M _ { d } ( k ) } \end{equation*}

and

\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k \in S } \frac { | g _ { 2 } ( k ) | } { M _ { d } ( k ) } \end{equation*}

(two-sided direct operator problems).

7) Given a domain $U$ and $d \in [ 0,3 ]$, find

\begin{equation*} \operatorname { inf } _ { z _ { 1 } , \ldots , z _ { n } \in U } \operatorname { max } _ { k \in S } \frac { \operatorname { Re } g _ { 1 } ( k ) } { M _ { d } ( k ) } \end{equation*}

and

\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k \in S } \frac { \operatorname { Re } g _ { 2 } ( k ) } { M _ { d } ( k ) }, \end{equation*}

where $g _ { 1 } ( k )$ and $g_2 ( k )$ are as above (one-sided conditional operator problems).

8) Given a finite set $S$ of integers, fixed complex numbers $b _ { j }$, $d \in [ 0,3 ]$, and two generalized power sums $g _ { 1 } ( k ) = \sum _ { j = 1 } ^ { n } b _ { j } ^ { \prime } ( k ) z _ { j } ^ { k }$, $g_2 ( k ) = \sum _ { j = 1 } ^ { n } b _ { j } ^ { \prime \prime } ( k ) z _ { j } ^ { k }$, how large can the quantities

\begin{equation*} \frac { | g _ { 1 } ( k ) | } { M _ { d ^ { \prime } } ( k ) } , \frac { | g _ { 2 } ( k ) | } { M _ { d ^ { \prime \prime } } ( k ) } \quad ( k \in S ) \end{equation*}

be made simultaneously depending only on $b _ { j }$, $d ^ { \prime }$, $d ^ { \prime \prime }$, $n$, and $S$ (simultaneous problems)?

9) Given two finite sets of integers $S _ { 1 }$ and $S _ { 2 }$, fixed complex numbers $b _ { j }$, $h ( m , k ) = \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k } w _ { j }^ { m }$, $| z _ { 1 } | \geq \ldots \geq | z _ { n } |$, $| w _ { 1 } | \geq \ldots \geq | w _ { n } |$, and $0 \leq d ^ { \prime } , d ^ { \prime \prime } \leq 3$, what is

\begin{equation*} \operatorname { inf } _ { z _ { j } , w _ { j } } \operatorname { max } _ { k \in S _ { 1 } , \atop m \in S _ { 2 } } \frac { | h ( m , k ) | } { M _ { d ^ { \prime } } ( k ) M _ { d^ { \prime \prime } } ( m ) } \end{equation*}

and what are the extremal systems (several variables problems)?

Turán and others obtained some lower bounds for some of the above problems.

Let $s _ { k } = z _ { 1 } ^ { k } + \ldots + z _ { n } ^ { k }$ be a pure power sum. Then

\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k = 1 , \ldots , n } \frac { | s _ { k } | } { M _ { 1 } ( k ) } = 1 \end{equation*}

and

\begin{equation*} \operatorname { inf } _ { z _ { j } } \operatorname { max } _ { k = 1 , \ldots , 2 n - 1 } \frac { | s _ { k } | } { M _ { 2 } ( k ) } = 1 \end{equation*}

(see also [a4]). These results were obtained in the equivalent form with $M _ { 1 } ( k ) = 1$ and $M _ { 2 } ( k ) = 1$, respectively.

Also, let $R _ { n } = \operatorname { min } _ { z _ { j } } \operatorname { max } _ { k = 1 , \ldots , n } | s _ { k } |$, where $\max| z _ { j } | = 1$. Then

\begin{equation} \tag{a2} R _ { n } > \frac { \operatorname { log } 2 } { 1 + \frac { 1 } { 2 } + \ldots + \frac { 1 } { n } }. \end{equation}

F.V. Atkinson [a2] improved this by showing that $R _ { n } > 1 / 5$. A. Biro [a3] proved that $R _ { n } > 1 / 2$ and that if $m > 0$ is such that $z _ { 1 } = \ldots = z _ { m } = 1$, $n \geq n _ { 0 }$, then

\begin{equation*} \operatorname { max } _ { j = 1 , \ldots , n - m + 1 } | s _ { j } | \geq m \left( \frac { 1 } { 2 } + \frac { m } { 8 n } + \frac { 3 m ^ { 2 } } { 64 n ^ { 2 } } \right). \end{equation*}

J. Anderson [a1] showed that if $\operatorname{min}_{j} | z _ { j } | = 1$, then $\operatorname { inf } _ { z _ { j } } \operatorname { max } _ { j = 1 , \ldots , n^2 } | s _ { j } | \geq \sqrt { n }$, and that if $n + 1$ is a prime number, then this lies in $[ \sqrt { n } , \sqrt { n + 1 } ]$; he also proved that if $m \in [ 1 , n - 1 ]$, then there exists a $c = c ( m )$ such that

\begin{equation*} \operatorname { max } _ { r = 1 , \ldots , c n } \frac { | z _ { 1 } ^ { r } + \ldots + z _ { n } ^ { r } | } { \operatorname { min } _ { k = 1 , \ldots , n } | z _ { k } ^ { r } | } \geq m. \end{equation*}

It is also known [a43] that, on the other hand, $R _ { n } < 1 - \operatorname { log } n / ( 3 n )$ for infinitely many $n$ and that $R _ { n } < 1 - 1 / ( 250 n )$ for large enough $n$.

P. Erdös proved that

\begin{equation*} M _ { 2 } = \operatorname { min } _ { z _ { j } } \operatorname { max } _ { k = 2 , \ldots , n + 1 } | s _ { k } | \leq 2 ( n + 1 ) ^ { 2 } e ^ { - \theta n }, \end{equation*}

where $\theta \approx 0.2784$ is the solution of the equation $x \operatorname { exp } ( x + 1 ) = 1$, and L. Erdös [a15] proved that if $n$ is large enough, then $\operatorname { exp } ( - 2 \theta n - 0.7823 \operatorname { log } n ) \leq M _ { 2 } \leq \operatorname { exp } ( - 2 \theta n + 4.5 \operatorname { log } n )$, where $\theta$ is the solution of the equation $1 + \theta + \operatorname { log } \theta = 0$.

E. Makai [a44] showed that

\begin{equation*} M _ { 3 } = \operatorname { min } _ { z _ { j } } \operatorname { max } _ { k = 3 , \ldots , n + 2 } | s _ { k } | < \frac { 1 } { 1.473 ^ { n } } \text { for } n > n _ { 0 }. \end{equation*}

For generalized power sums $g ( k ) = \sum _ { j = 1 } ^ { n } b _ { j } z _ { j } ^ { k }$, Turán proved that if $\min_{ z _ { j }} | z _ { j } | = 1$, then

\begin{equation*} \operatorname { max } _ { k = m + 1 , \ldots , m + n } | g ( k ) | \geq \left( \frac { n } { 2 e ( m + n ) } \right) ^ { n } | b _ { 1 } + \ldots + b _ { n } |. \end{equation*}

Makai [a45] and N.G. de Bruijn [a4] proved, independently, that $( n / ( 2 e ( m + n ) ) ) ^ { n }$ can be replaced with $1 / P _ { m , n }$, where $P _ { m , n } = \sum _ { j = 0 } ^ { n - 1 } \left( \begin{array} { c } { m + j } \\ { j } \end{array} \right) 2 ^ { j }$. If, however, one replaces it with $1 / ( P _ { m ,\, n } - \epsilon )$ for any $\epsilon > 0$, then the above inequality fails. Turán also proved that if $\operatorname{min}_{j} | z _ { j } | = 1$, then

\begin{equation*} \operatorname { max } _ { k = m + 1 , \ldots , m + n } | g ( k ) | \geq \frac { 1 } { 3 } | g ( 0 ) | \prod _ { j = 1 } ^ { n } \frac { | z _ { j } | - \operatorname { exp } ( - 1 / m ) } { | z _ { j } | + 1 }. \end{equation*}

G. Halasz showed that for any $k > 1$,

S. Gonek [a18] proved that for all $r > 0$,

\begin{equation*} \operatorname { max } _ { 1 \leq k \leq 4 \left( \begin{array} { c } { n + r - 1 } \\ { r } \end{array} \right)} | g ( k ) | \geq | g ( 0 ) | \left( 2 e \left( \begin{array} { c } { n + r - 1 } \\ { r } \end{array} \right) \right) ^ { - 1 / r }. \end{equation*}

In the case of the maximum norm, V. Sos and Turán [a46] obtained the following result. Let $1 = | z _ { 1 } | \geq \ldots \geq | z _ { n } |$. Then for any integer $m \geq 0$,

\begin{equation*} \operatorname { max } _ { k = m + 1 , \ldots , m + n } | g ( k ) | \geq c _ { m , n } , \operatorname { min } _ { j = 1 , \ldots , n } | b _ { 1 } + \ldots + b _ { j } | \end{equation*}

with $c _ { m , n } = 2 ( n / ( 8 e ( m + n ) ) ) ^ { n }$. G. Kolesnik and E.G. Straus [a42] improved this by showing that one can take $c _ { m , n } = \sqrt { n } ( n / ( 4 e ( m + n ) ) ) ^ { n }$. On the other hand, Makai [a45] showed that for

the inequality fails for some $m$ and $z_j$.

Considering different ranges for $k$, Halasz [a19] proved that if $m , n < N$, then

\begin{equation*} \operatorname { min } _ { k = m + 1 , \ldots , m + N } | g ( k ) | \geq \end{equation*}

\begin{equation*} \geq \frac { n } { 4 N ^ { 3 / 2} } \operatorname { exp } \left( - 30 n \left( \frac { 1 } { \operatorname { log } ( N / n ) } + \frac { 1 } { \operatorname { log } ( N / m ) } \right) \right) \times \times \operatorname { min } _ { l \leq n } \left| \sum _ { j = 1 } ^ { l } b _ { j }\right| . \end{equation*}

Other norms and conditions.

The following results are obtained for two-sided problems with other norms and conditions.

A) ([a17], [a47], [a8], [a45]). Let $z_j$ be ordered so that $0 = | z _ { 1 } - 1 | \leq \ldots \leq | z _ { n } - 1 |$. Assume that $m \geq - 1$ and $n > 1$. Then

\begin{equation*} \operatorname { max } _ { k = m + 1 , \ldots , m + n } | g ( k ) | \geq \end{equation*}

\begin{equation*} \geq \frac { 1 } { 8 } \left( \frac { n - 1 } { 8 e ( m + n ) } \right) ^ { n } \operatorname { min }_ j | b _ { 1 } + \ldots + b _ { j } |. \end{equation*}

B) ([a91]). Let $z_j$ be ordered as in A). Assume that $m > - 1$ and $0 < \delta _ { 1 } < \delta _ { 2 } < n / ( m + n + 1 )$, let $h$ be the largest integer satisfying $| 1 - z _ { h } | < \delta _ { 1 }$ and let $l$ be the smallest integer satisfying $| 1 - z _{l + 1} | > \delta _ { 2 }$ (if such an integer does not exist, take $l = n$). Then

\begin{equation*} \operatorname { max } _ { k = m + 1 , \ldots , m + n } | g ( k ) | \geq \end{equation*}

\begin{equation*} \geq 2 \left( \frac { \delta _ { 1 } - \delta _ { 2 } } { 12 e } \right) ^ { n } \operatorname { min } _ { j = h , \ldots , l } | b _ { 1 } + \ldots + b _ { j } |. \end{equation*}

C) ([a12]). Let $m \geq 0$ and let $k$, $k_{1}$, $k_2$ be such that

\begin{equation*} | z _ { 1 } | \geq \ldots \geq | z _ { k _ { 1 } } | > \frac { m + 2 n } { m + n } \geq \end{equation*}

\begin{equation*} \geq | z _ { k_1 } + 1 | \geq \ldots \geq | z _ { k } | = 1 \geq \ldots \geq | z _ { k _ { 2 } } - 1 | > > \frac { m } { m + n } \geq | z _ { k _ { 2 } } | \geq \ldots \geq | z _ { n } |. \end{equation*}

Then

\begin{equation*} \operatorname { max } _ { r = m + 1 , \ldots , m + n } | g ( r ) | \geq \end{equation*}

\begin{equation*} \geq \frac { 1 } { n } \left( \frac { n } { 16 e ( m + n ) } \right) ^ { n } \times \times \operatorname{min} _ { k _ { 1 } \leq l _ { 1 } \leq k \leq l _ { 2 } \leq k _ { 2 } } | b _ {l_{ 1} } + \ldots + b _ {l_{ 2 }} |. \end{equation*}

D) ([a59]). If $m > - 1$ and $\operatorname {min}_{ \mu \neq \nu} | z _ { \mu } - z _ { \nu } | \geq \delta \operatorname { max } _ { j } | z _ { j }|$, then

\begin{equation*} \frac { \max_{k = m + 1 , \ldots , m + n}| g ( k ) | } { \sum _ { j = 1 } ^ { n } | b _ { j } z _ { j } ^ { k } | } \geq \frac { 1 } { n } ( \frac { \delta } { 2 } ) ^ { n - 1 }. \end{equation*}

E) ([a8]). If $m > - 1$ and $r$ is such that $\operatorname{min}_{j \neq r} | z _ { j } - z _ { r } | \geq \delta | z _ { r } |$, then there exists a $ k \in [ m + 1 , m + n ]$ such that

\begin{equation*} | g ( k ) | \geq ( \frac { \delta } { 2 + 2 \delta } ) ^ { n - 1 } | b _ { r } z _ { r} ^ { k } |. \end{equation*}

F) ((Halasz). Let $m_1$ and $m _ { 2 }$ be non-negative integers, $m = \operatorname { max } ( m _ { 1 } , m _ { 2 } )$, and $S = [ - m _ { 1 } - n , - m _ { 1 } - 1 ] \cup [ m _ { 2 } + 1 , m _ { 2 } + n ]$. Assume that $z _ { 1 } \dots z _ { n } \neq 0$. Then there exists an integer $k \in S$ such that

\begin{equation*} | g ( k ) | \geq \left( \frac { n } { 8 e ( m + n ) } \right) ^ { n } | g ( 0 ) |. \end{equation*}

G) (Turán). If $S = [ m + 1 , m + n ] \cup [ 2 m + 1,2 m + n ]$, then the above inequality holds with $6$ instead of $8$.

Problems of type 3) and 7).

Assume that $\kappa \leq | \operatorname { arc } z _ { j } | \leq \pi$, $j = 1 , \ldots , n$, with $0 < \kappa \leq \pi / 2$, let $a _ { j }$ be real numbers, and let $\phi ( z ) = z ^ { k } + a _ { 1 } z ^ { k - 1 } + \ldots + a _ { k } \neq 0$ for $| z | > \rho \in ( 0,1 )$. Define $G _ { 2 } ( k ) = \sum _ { j = 1 } ^ { n } b _ { j } \phi ( z _ { j } ) z _ { j } ^ { k }$ for some fixed complex numbers $b _ { j }$. Assuming that $\operatorname{min}_{j} | z _ { j } | = 1$, Turán proved that $\max _ r \operatorname { Re } G _ { 2 } ( r ) \geq A$ and $\operatorname { min}_r \operatorname { Re } G _ { 2 } ( r ) \leq - A$, where

\begin{equation*} A = \frac { 1 } { 6 n 16 ^ { n } } \left( \frac { 1 + \rho } { 2 } \right) ^ { m } \left( \frac { 1 - \rho } { 2 } \right) ^ { 2 n + k } \left| \operatorname { Re } \sum _ { j = 1 } ^ { n } b _ { j } \right| \end{equation*}

and the minimum is taken over all integers $r \in [ m + 1 , m + n ( 3 + \pi / k ) ]$.

If $\phi ( z ) = 1$, then the above inequalities hold with

\begin{equation*} A = \frac { 1 } { 6 n } \operatorname { min } _ { n \leq x \leq 2 n } \left( \frac { x } { 4 e ( m + x ) } \right) ^ { x } \left| \operatorname { Re } \sum _ { j = 1 } ^ { n } b _ { j } \right|. \end{equation*}

Also, if $P _ { j } ( x )$ are polynomials of degree $k_j - 1$, $G _ { 1 } ( r ) = \sum _ { j = 1 } ^ { n } P _ { j } ( r ) z _ { j } ^ { r }$ and $K = k _ { 1 } + \ldots + k _ { n }$, then $\operatorname {max}_{r}\operatorname { Re } G _ { 1 } ( r ) \geq B$ and $\min_r \operatorname{Re} G _ { 1 } ( r ) \leq - B$, where

\begin{equation*} B = \frac { 1 } { 6 K } \left( \frac { K } { 4 e ( m + 2 K ) } \right) ^ { 2 K } \left| \operatorname { Re } \sum _ { j = 0 } ^ { n } P _ { j } ( 0 ) \right| \end{equation*}

and the range of $r$ is $[ m + 1 , m + K ( 3 + \pi / \kappa ) ]$.

Assume now that $\operatorname {max}_{j} | z _ { j } | = 1$. Let $G _ { 2 } ( r )$ be as defined above, and assume $\phi ( z ) \neq 0$ for $z \in \{ | z | \geq \rho \} \cup \{ | \operatorname { arc } z | < \kappa \}$, where $0 < \rho < 1$ and $0 < \kappa < \pi / 2$. Assume also that $\kappa \leq | \operatorname { arc } z _ { j } | < \pi$, $j = 1 , \ldots , n$. Take any $\delta _ { 1 }$, $\delta _ { 2 }$ satisfying $1 > \delta _ { 1 } > \delta _ { 2 } \geq \rho$ and define $h_{1}$, $h _ { 2 }$ by

\begin{equation*} 1 = | z _ { 1 } | \geq \ldots \geq | z _ { h _ { 1 } } | \geq \delta _ { 1 } > \end{equation*}

\begin{equation*} > | z _ { h _ { 1 } } + 1 | \geq \ldots \geq | z _ { h _ { 2 } } | > \delta _ { 2 } \geq \end{equation*}

\begin{equation*} \geq | z _ { h_2 } + 1 | \geq \ldots \geq | z _ { n } |. \end{equation*}

(If $h_{1}$ or $h _ { 2 }$ do not exist, replace them with $n$.) Put $I = [ m + 1 , m + ( n + k ) ( 3 + \pi / k ) ]$ and

\begin{equation*} M = \frac { 1 } { 3 ( n + k ) } \left( \frac { \delta _ { 1 } - \delta _ { 2 } } { 16 } \right) ^ { 2 n + 2 k } \delta _ { 2 } ^ { m + ( n + k ) ( 1 + \pi / k ) }\times \end{equation*}

\begin{equation*} \times \operatorname { min } _ { h _ { 1 } \leq j \leq h _ { 2 } } | \operatorname { Re } ( b _ { 1 } + \ldots + b _ { j } ) |. \end{equation*}

Then $\max_{r \in I} \text{Re} \, G_2 (r ) \geq M$ and $\operatorname{min}_{r\in I} \operatorname{Re} G _ { 2 } ( r ) \leq - M$. If $\phi ( z ) = 1$, then the above result holds with $k = \rho = 0$.

J.D. Buchholtz [a5], [a6] proved that if $\operatorname {max}_{j} | z _ { j } | = 1$, then

\begin{equation*} \operatorname { max } _ { k = 1 , \ldots , n } \left( \frac { 1 } { n } | s _ { k } | \right) ^ { 1 / k } > \frac { 1 } { 5 } > \frac { 1 } { 2 + \sqrt { 8 } }, \end{equation*}

respectively, where the last result is the best possible.

R. Tijdeman [a47] proved the following result for "operator-type problems" .

Let $P _ { j } ( x )$ be fixed complex polynomials of degree $k_j - 1$ and let $G _ { 1 } ( k ) = \sum _ { j = 1 } ^ { n } P _ { j } ( k ) z _ { j } ^ { k }$. Then for every integer $m \geq 0$, $K = k _ { 1 } + \ldots + k _ { n }$, and $\operatorname{min}_{j} | z _ { j } | = 1$, the inequality

\begin{equation} \tag{a3} \operatorname { min } _ { r = m + 1 , \ldots , m + K } | G _ { 1 } ( r ) | \geq \frac { 1 } { P _ { m , K } } \left| \sum _ { j = 1 } ^ { n } P _ { j } ( 0 ) \right| \end{equation}

holds, where $P _ { m , K }$ is defined above and the factor $1 / P _ { m , K}$ is the best possible; also, if $\operatorname {max}_{j} | z _ { j } | = 1$, then (a3) holds with $( K / ( 8 e ( m + K ) ) ) ^ { K }$ instead of $1 / P _ { m , K}$.

J. Geysel [a17] improved the above constant to

\begin{equation*} \frac { 1 } { 4 } \left( \frac { K - 1 } { 8 e ( m + K ) } \right) ^ { K }. \end{equation*}

Turán studied the other "operator-type problem" for $G _ { 2 } ( r ) = \sum _ { j = 1 } ^ { n } b _ { j } \phi ( z _ { j } ) z _ { j } ^ { k }$. Let $b _ { j }$ be fixed complex numbers and let $\phi ( z ) = z ^ { k } + a _ { 1 } z ^ { k - 1 } + \ldots + a _ { k }$ be a polynomial with no zeros outside $| z | < \rho$. Assume that $m > - 1$, $0 < \rho < 1$ and $\operatorname {max}_{j} | z _ { j } | = 1$. Then

(a4)

with

\begin{equation*} c _ { m , n } = 2 ^ { - n } \left( \frac { 1 + \rho } { 2 } \right) ^ { m } \left( \frac { 1 - \rho } { 2 } \right) ^ { n + k }. \end{equation*}

In case of the maximum norm and $1 = | z _ { 1 } | \geq \ldots \geq | z _ { n } | > 0$, Turán proved (a4) with

\begin{equation*} c _ { m ,\, n } = \left\{ \begin{array} { l l } { 2 ^ { 1 - n } \left( \frac { n + k } { 4 e ( m + n + k ) } \right) ^ { n + k } } & { \text { if } \frac { m } { m + n + k } \geq \rho, } \\ { \rho ^ { m } 2 ^ { 1 - n } \left( \frac { 1 - \rho } { 4 } \right) ^ { n + k } } & { \text { if } \frac { m } { m + n + k } < \rho. } \end{array} \right. \end{equation*}

He also proved the following "simultaneous problem" . Let $\operatorname{min}_j | z _ { j } | = \operatorname { min } _ { j } | w _ { j } | = 1$. For any integers $m > - 1$ and $n _ { 1 } , n _ { 2 } \geq 1$ there exist a $ k \in [ m + 1 , m + n _ { 1 } n _ { 2 } ]$ such that the inequalities

and

hold simultaneously.

References

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[a68] P. Turan, "Uber lakunaren Potenzreihen" Rev. Math. Pures Appl. , I (1956) pp. 27–32
[a69] P. Turan, "On the so-called density hypothesis of zeta-function of Riemann" Acta Arith. , IV : 1 (1958) pp. 31–56
[a70] P. Turan, "Zur Theorie der Dirichletschen Reihen" Euler Festschr. (1959) pp. 322–336
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How to Cite This Entry:
Turán theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tur%C3%A1n_theory&oldid=50037
This article was adapted from an original article by Grigori Kolesnik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article