Comparison function

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