Exhaustion by balls and entire functions of bounded $\mathbf{L}$-index in joint variables

Abstract

The criteria describe local behavior of partial derivatives of entire function on sphere in an-dimensional complex space. Our main result provides an upper bound for maximal absolute value of partial derivatives of entire function on the sphere in terms of the absolute value of the function at the center of the sphere multiplied by some constant. This constant depends only on the radius of sphere and is independent of the location of its center. Some of the obtained results are new even for entire functions with a bounded index in joint variables, i.e., L() ≡ 1, because we use an exhaustion of C by balls instead an exhaustion of C by polydiscs. The ball exhaustion is based on Cauchy's integral formula for a ball. Also we weaken sufficient conditions of index boundedness in our main result by replacing an universal quantifier by an existential quantifier. The polydisc analogues of the obtained results are fundamental in theory of entire functions of bounded index in joint variables. They are used for estimating the maximal absolute value by the minimal absolute value, for estimating partial logarithmic derivatives and distribution of zeroes.

Functions analytic in a unit ball of bounded L-index in joint variables

“…In this paper we will introduce and study analytic in a ball functions of bounded L-index in joint variables. Of course, there are wide bibliography about entire functions of bounded L-index in joint variables [19]- [22], [29]- [33], [40]- [41].…”

On Existence of Main Polynomial for Analytic Vector-Valued Functions of Bounded L-Index in the Unit Ball

“…We obtain a p+1 r p+1 0 > aν 0 r p+1 0 η p+1 ≥ a ν 0 r ν 0 . The last inequality contradicts (14). Thus, the vector-function F has bounded L-index in joint variables.…”

“…The concept of boundedness of L-index in joint variables were considered for other classes of analytic functions. They are differed domains of analyticity: the unit ball [5,4,12,13], the polydisc [7,11], the Cartesian product of the unit disc and complex plane [8], n-dimensional complex space [12,14]. Vector-valued functions of one and several complex variables having bounded index were considered in [16,18,15,20,19,17].…”

Growth Estimates for Analytic Vector-Valued Functions in the Unit Ball Having Bounded $\mathbf{L}$-index in Joint Variables

“…The concept of boundedness of L-index in joint variables were considered for other classes of analytic functions. They are differed domains of analyticity: the unit ball [5,4,11,13], the polydisc [8,10], the Cartesian product of the unit disc and complex plane [9], n-dimensional complex space [7,11,14]. Vector-valued functions of one and several complex variables having bounded index were considered in [18,20,17,23,21,19].…”