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

Andriy Bandura, Oleh Skaskiv

In this paper growth estimates of entire in $\mathbb{C}^n$ function of bounded $\mathbf{L}$-index in joint variables are obtained. They describe the behavior of maximum modulus of an entire function on a skeleton in a polydisc by behavior of the function $\mathbf{L}(z)=(l_1(z),\ldots,l_n(z)),$ where for every $j\in\{1,\ldots, n\}$ $l_j:\mathbb{C}^n\to \mathbb{R}_+$ is a continuous function. We generalized known results of W. K. Hayman, M. M. Sheremeta, A. D. Kuzyk and M. T. Borduyak to 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.