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A function that is used in studying the character of growth of the modulus of an [[Entire function|entire function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c0236201.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c0236202.png" />; a comparison is normally made here between the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c0236203.png" /> and that of a certain  "good"  entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c0236204.png" />. Here naturally arises the problem of describing a sufficiently broad set of entire functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c0236205.png" /> the elements of which could successfully be used as  "comparison standards" .
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An entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c0236206.png" /> is called a comparison function, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c0236207.png" />, if: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c0236208.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c0236209.png" />); and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362010.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362011.png" />. An entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362012.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362014.png" />-comparable if there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362016.png" />, such that
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/c/c023/c023620/c02362017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A function that is used in studying the character of growth of the modulus of an [[Entire function|entire function]]  $  a ( z) $
 +
when  $  z \rightarrow \infty $;  
 +
a comparison is normally made here between the behaviour of  $  | a ( z) | $
 +
and that of a certain  "good" entire function  $  A ( z) $.
 +
Here naturally arises the problem of describing a sufficiently broad set of entire functions  $  \mathfrak A = \{ A ( z) \} $
 +
the elements of which could successfully be used as  "comparison standards" .
  
The lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362018.png" /> of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362019.png" /> for which the relation (1) is fulfilled is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362021.png" />-type of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362022.png" />-comparable entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362023.png" />. The following theorem on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362024.png" />-types holds: If an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362025.png" /> is comparable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362027.png" />, then its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362028.png" />-type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362029.png" /> can be calculated using the formula
+
An entire function  $  A ( z) = \sum _ {k = 0 }  ^  \infty  A _ {k} z  ^ {k} $
 +
is called a comparison function, or  $  A ( z) \in \mathfrak A $,
 +
if: 1) $  A _ {k} > 0 $(
 +
$  k = 0, 1 , . . . $);
 +
and 2)  $  A _ {k + 1 }  /A _ {k} \downarrow 0 $
 +
as  $  k \rightarrow \infty $.  
 +
An entire function $  a ( z) $
 +
is called  $  A $-
 +
comparable if there exists a constant  $  \tau $,
 +
$  \tau > 0 $,  
 +
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/c/c023/c023620/c02362030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
a ( z)  = O ( A ( \tau  | z | )) \ \
 +
\textrm{ as }  z \rightarrow \infty .
 +
$$
  
The given class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362031.png" /> of comparison functions is known to give a complete solution of this problem, since for any entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362032.png" />, other than a polynomial, there exists a comparison function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362034.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362035.png" /> is comparable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362036.png" /> and such that its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362037.png" />-type is equal to 1.
+
The lower bound  $  \sigma $
 +
of the numbers  $  \{ \tau \} $
 +
for which the relation (1) is fulfilled is called the  $  A $-
 +
type of the  $  A $-
 +
comparable entire function a ( z) $.  
 +
The following theorem on  $  A $-
 +
types holds: If an entire function  $  a ( z) = \sum _ {k = 0 }  ^  \infty  a _ {k} z  ^ {k} $
 +
is comparable with $  A ( z) $,
 +
$  A ( z) \in \mathfrak A $,
 +
then its $  A $-
 +
type $  \sigma $
 +
can be calculated using the formula
  
If an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362038.png" /> is comparable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362040.png" />, and its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362041.png" />-type is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362042.png" />, then the function
+
$$ \tag{2 }
 +
\sigma  = \
 +
\lim\limits _ {k \rightarrow \infty } \
 +
\sup  \left |
 +
\frac{a _ {k} }{A _ {k} }
 +
\right |  ^ {1/k} .
 +
$$
  
<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/c/c023/c023620/c02362043.png" /></td> </tr></table>
+
The given class $  \mathfrak A $
 +
of comparison functions is known to give a complete solution of this problem, since for any entire function  $  a ( z) $,
 +
other than a polynomial, there exists a comparison function  $  A ( z) $,
 +
$  A ( z) \in \mathfrak A $,
 +
such that  $  a ( z) $
 +
is comparable with  $  A ( z) $
 +
and such that its  $  A $-
 +
type is equal to 1.
  
is analytic, according to (2), for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362044.png" />; it is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362046.png" />-associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362047.png" />. In this case, the generalized Borel representation holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362048.png" />:
+
If an entire function  $  a ( z) = \sum _ {k = 0 }  ^  \infty  a _ {k} z  ^ {k} $
 +
is comparable with  $  A ( z) $,
 +
$  A ( z) \in \mathfrak A $,
 +
and its  $  A $-
 +
type is equal to  $  \sigma $,  
 +
then the function
  
<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/c/c023/c023620/c02362049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$
 +
\gamma _ {A} ( t)  = \
 +
\sum _ {k = 0 } ^  \infty 
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362050.png" /> is taken as a comparison function, then (3) is the classical Borel integral representation of entire functions of exponential type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362051.png" />.
+
\frac{a _ {k} /A _ {k} }{t ^ {k + 1 } }
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362052.png" /> holds in (3), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362053.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362054.png" />) is a Mittag-Leffler function, then (3) is an integral representation for any entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362055.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362056.png" /> and of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362057.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362058.png" /> is the type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362059.png" /> in the classical sense).
+
$$
  
For certain cases of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362060.png" />, an inverse transformation to (3) has been constructed (see, for example [[#References|[1]]], which has a bibliography relating to comparison functions). Comparison functions and the Borel representation (3) are used in various questions of analysis (see, for example, [[#References|[2]]], [[#References|[3]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362061.png" /> is the class of entire functions that are comparable with a given comparison function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362062.png" />, then, for any sequence of comparison functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362063.png" />, there always exists an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362064.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023620/c02362065.png" />.
+
is analytic, according to (2), for  $  | t | > \sigma $;
 +
it is called  $  A $-
 +
associated with  $  a ( z) $.
 +
In this case, the generalized Borel representation holds for  $  a ( z) $:
 +
 
 +
$$ \tag{3 }
 +
a ( z)  = \
 +
 
 +
\frac{1}{2 \pi i }
 +
 
 +
\int\limits _ {| t | = \sigma + \epsilon }
 +
A ( zt) \gamma _ {A} ( t)  dt \ \
 +
( \forall \epsilon :  \epsilon > 0).
 +
$$
 +
 
 +
If  $  A ( z) \equiv e  ^ {z} $
 +
is taken as a comparison function, then (3) is the classical Borel integral representation of entire functions of exponential type  $  \sigma $.
 +
 
 +
If  $  A ( z) \equiv E _  \rho  ( z) $
 +
holds in (3), where  $  E _  \rho  ( z) = \sum _ {k = 0 }  ^  \infty  z  ^ {k} / \Gamma ( 1 + k/ \rho ) $(
 +
$  \rho > 0 $)
 +
is a Mittag-Leffler function, then (3) is an integral representation for any entire function  $  a ( z) $
 +
of order  $  \rho $
 +
and of type  $  \sigma ^ {1/ \rho } $(
 +
$  \sigma ^ {1/ \rho } $
 +
is the type of  $  a ( z) $
 +
in the classical sense).
 +
 
 +
For certain cases of  $  A ( z) $,  
 +
an inverse transformation to (3) has been constructed (see, for example [[#References|[1]]], which has a bibliography relating to comparison functions). Comparison functions and the Borel representation (3) are used in various questions of analysis (see, for example, [[#References|[2]]], [[#References|[3]]]). If $  [ A;  \infty ) $
 +
is the class of entire functions that are comparable with a given comparison function $  A ( z) $,  
 +
then, for any sequence of comparison functions $  \{ A _ {n} \} _ {n = 0 }  ^  \infty  $,  
 +
there always exists an entire function $  a ( z) $
 +
such that $  a ( z) \notin \cup _ {n = 0 }  ^  \infty  [ A _ {n} ;  \infty ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.P. Boas,  R.C. Buck,  "Polynomial expansions of analytic functions" , Springer &amp; Acad. Press (U.S.A. &amp; Canada)  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.M. Dzhrbashyan,  "Integral transforms and representation of functions in the complex domain" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.A. Kaz'min,  "A certain problem of A.O. Gel'fond"  ''Math. USSR Sb.'' , '''19''' :  4  (1973)  pp. 509–530  ''Mat. Sb.'' , '''90''' :  4  (1973)  pp. 521–543</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.P. Boas,  R.C. Buck,  "Polynomial expansions of analytic functions" , Springer &amp; Acad. Press (U.S.A. &amp; Canada)  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.M. Dzhrbashyan,  "Integral transforms and representation of functions in the complex domain" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.A. Kaz'min,  "A certain problem of A.O. Gel'fond"  ''Math. USSR Sb.'' , '''19''' :  4  (1973)  pp. 509–530  ''Mat. Sb.'' , '''90''' :  4  (1973)  pp. 521–543</TD></TR></table>

Revision as of 17:45, 4 June 2020


A function that is used in studying the character of growth of the modulus of an entire function $ a ( z) $ when $ z \rightarrow \infty $; a comparison is normally made here between the behaviour of $ | a ( z) | $ and that of a certain "good" entire function $ A ( z) $. Here naturally arises the problem of describing a sufficiently broad set of entire functions $ \mathfrak A = \{ A ( z) \} $ the elements of which could successfully be used as "comparison standards" .

An entire function $ A ( z) = \sum _ {k = 0 } ^ \infty A _ {k} z ^ {k} $ is called a comparison function, or $ A ( z) \in \mathfrak A $, if: 1) $ A _ {k} > 0 $( $ k = 0, 1 , . . . $); and 2) $ A _ {k + 1 } /A _ {k} \downarrow 0 $ as $ k \rightarrow \infty $. An entire function $ a ( z) $ is called $ A $- comparable if there exists a constant $ \tau $, $ \tau > 0 $, such that

$$ \tag{1 } a ( z) = O ( A ( \tau | z | )) \ \ \textrm{ as } z \rightarrow \infty . $$

The lower bound $ \sigma $ of the numbers $ \{ \tau \} $ for which the relation (1) is fulfilled is called the $ A $- type of the $ A $- comparable entire function $ a ( z) $. The following theorem on $ A $- types holds: If an entire function $ a ( z) = \sum _ {k = 0 } ^ \infty a _ {k} z ^ {k} $ is comparable with $ A ( z) $, $ A ( z) \in \mathfrak A $, then its $ A $- type $ \sigma $ can be calculated using the formula

$$ \tag{2 } \sigma = \ \lim\limits _ {k \rightarrow \infty } \ \sup \left | \frac{a _ {k} }{A _ {k} } \right | ^ {1/k} . $$

The given class $ \mathfrak A $ of comparison functions is known to give a complete solution of this problem, since for any entire function $ a ( z) $, other than a polynomial, there exists a comparison function $ A ( z) $, $ A ( z) \in \mathfrak A $, such that $ a ( z) $ is comparable with $ A ( z) $ and such that its $ A $- type is equal to 1.

If an entire function $ a ( z) = \sum _ {k = 0 } ^ \infty a _ {k} z ^ {k} $ is comparable with $ A ( z) $, $ A ( z) \in \mathfrak A $, and its $ A $- type is equal to $ \sigma $, then the function

$$ \gamma _ {A} ( t) = \ \sum _ {k = 0 } ^ \infty \frac{a _ {k} /A _ {k} }{t ^ {k + 1 } } $$

is analytic, according to (2), for $ | t | > \sigma $; it is called $ A $- associated with $ a ( z) $. In this case, the generalized Borel representation holds for $ a ( z) $:

$$ \tag{3 } a ( z) = \ \frac{1}{2 \pi i } \int\limits _ {| t | = \sigma + \epsilon } A ( zt) \gamma _ {A} ( t) dt \ \ ( \forall \epsilon : \epsilon > 0). $$

If $ A ( z) \equiv e ^ {z} $ is taken as a comparison function, then (3) is the classical Borel integral representation of entire functions of exponential type $ \sigma $.

If $ A ( z) \equiv E _ \rho ( z) $ holds in (3), where $ E _ \rho ( z) = \sum _ {k = 0 } ^ \infty z ^ {k} / \Gamma ( 1 + k/ \rho ) $( $ \rho > 0 $) is a Mittag-Leffler function, then (3) is an integral representation for any entire function $ a ( z) $ of order $ \rho $ and of type $ \sigma ^ {1/ \rho } $( $ \sigma ^ {1/ \rho } $ is the type of $ a ( z) $ in the classical sense).

For certain cases of $ A ( z) $, an inverse transformation to (3) has been constructed (see, for example [1], which has a bibliography relating to comparison functions). Comparison functions and the Borel representation (3) are used in various questions of analysis (see, for example, [2], [3]). If $ [ A; \infty ) $ is the class of entire functions that are comparable with a given comparison function $ A ( z) $, then, for any sequence of comparison functions $ \{ A _ {n} \} _ {n = 0 } ^ \infty $, there always exists an entire function $ a ( z) $ such that $ a ( z) \notin \cup _ {n = 0 } ^ \infty [ A _ {n} ; \infty ) $.

References

[1] R.P. Boas, R.C. Buck, "Polynomial expansions of analytic functions" , Springer & Acad. Press (U.S.A. & Canada) (1958)
[2] M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)
[3] Yu.A. Kaz'min, "A certain problem of A.O. Gel'fond" Math. USSR Sb. , 19 : 4 (1973) pp. 509–530 Mat. Sb. , 90 : 4 (1973) pp. 521–543
How to Cite This Entry:
Comparison function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Comparison_function&oldid=18614
This article was adapted from an original article by Yu.A. Kaz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article