Asymptotic estimates of entire functions of bounded $\mathbf{L}$-index in joint variables

Abstract

In this paper, there are obtained growth estimates of entire in ${\mathbb{C}}^n$ function of bounded $\mathbf{L}$-index in joint variables. They describe the behaviour of maximum modulus of entire function on a skeleton in a polydisc by behaviour of the function $\mathbf{L}(z)=(l_1(z),\dots ,l_n(z)),$ where for every $j\in \{1,\dots ,n\}$ \ $l_j:{\mathbb{C}}^n\to {\mathbb{R}}_{+}$ is a continuous function. We generalised known results of W. K. Hayman, M. M. Sheremeta, A. D. Kuzyk, M. T. Borduyak, T. O. Banakh and V. O. Kushnir for a wider class of functions $\mathbf{L}.$ One of our estimates is sharper even for entire in $\mathbb{C}$ functions of bounded $l$-index than Sheremeta's estimate.

11 pages