Entire bivariate functions of exponential type

In this paper we will analysis the concepts of bivariate entire complex valued functions of exponential type. To accomplish this goal, we begin with the presentation of a notion of bounded index for bivariate complex functions. Using this notion we present a series of sufficient conditions that ensure that exponential type is preserved.

| f (z)| < const.e α|z| everywhere. The infimum of the set of these α is called the type of f (z).
If f (z, w) is a bivariate entire function in the bicylinder {(z, w) ∈ C 2 : |z − a| < r 1 , |w − b| < r 2 } then at point (a, b), f (z, w) have a bivariate Taylor expansion Similar to Gross [5] we presented in [9] the following notion of bounded index of bivariate entire function.

Definition 1
A bivariate entire function f is said to be of bounded index provided that there exist integers M and N independent of z and w such that max 0≤k≤M;0≤l≤N for all i = 0, 1, 2, . . . and j = 0, 1, 2, . . . and all z and w.
We shall say that f is of index (M, N ) if N ad M are the smallest integers for which above inequality holds. A bivariate entire function which is not of bounded index is said to be of unbounded index. One should observe that a f bivariate entire function is of bounded index then there exist integers M ≥ 0, N ≥ 0 and some C > 0, where i, j = 0, 1, 2, 3, . . ..

Slight variation of (1.1) is following
We also consider those variation of (1.2) obtained by replacing where p is any positive integer. We shall show in the sequel that entire bivariate functions satisfying inequality (1.2) or any of the conditions obtained from it by the substitutions suggested above are bivariate functions of exponential type.

Main result
Let f (z, w) be a bivariate function. Suppose that 0 < ρ < ∞ and let as define The functions which satisfy the above equality are said to be functions of exponential type τ . We can state the classical Borel [1] lemma for bivariate functions as follows: Lemma 1 Let T be a continuous nondecreasing function on [r 0 , ∞) × [s 0 , ∞) for some r 0 and s 0 such that T (r 0 , s 0 ) ≥ 1. Then for all r 1 and r 2 outside a possible exceptional set E whose measure is at most 4, that is,  r 2 ) is the maximum modulus of f (i, j) on |z| = r 1 and |w| = r 2 , that is

then f is of exponential type.
Proof For convenience we choose C = 1. The proof of other C is similar. For any arbitrary bivariate entire function F and any complex numbers A and B, we have Let m and n any integers, a, b, ξ, ζ complex numbers with |ξ | = 1 and |ζ | = 1.
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