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

Abstract

For entire functions of several complex variables, we prove criteria of boundedness of $\mathbf{L}$-index in joint variables. Here $\mathbf{L}: \mathbb{C}^n\to\mathbb{R}^n_+$ is a continuous vector function. The criteria describe local behavior of partial derivatives of entire function on sphere in an $n$-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., $\mathbf{L}(z)\equiv 1,$ because we use an exhaustion of $\mathbb{C}^n$ by balls instead an exhaustion of $\mathbb{C}^n$ 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.