# Comparison function

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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=51263
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