Some Results on Composition of Analytic Functions in a Unit Polydisc

Abstract

The manuscript is an attempt to consider all methods which are applicable to investigation a directional index for composition of an analytic function in some domain and an entire function. The approaches are applied to find sufficient conditions of the $L$-index boundedness in a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$, where the continuous function $L$ satisfies some growth condition and the condition of positivity in the unit polydisc. The investigation is based on a counterpart of the Hayman Theorem for the class of analytic functions in the polydisc and a counterpart of logarithmic criterion describing local conduct of logarithmic derivative modulus outside some neighborhoods of zeros. The established results are new advances for the functions analytic in the polydisc and in multidimensional value distribution theory.