# Turán theory

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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 $d \in [ 0,3 ]$,

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

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

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

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

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

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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How to Cite This Entry:
Turán theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tur%C3%A1n_theory&oldid=55757
This article was adapted from an original article by Grigori Kolesnik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article