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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" .

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 [[Function of exponential type|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====
<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>