Functions analytic in the unit ball having bounded $L$-index in a direction

Abstract

We propose a generalization of a concept of bounded index for analytic functions in the unit ball. Use of directional derivatives gives us a possibility to deduce the necessary and sufficient conditions of boundedness of $L$-index in a direction for analytic functions of several variables, namely, we obtain an analog of Hayman's theorem and a logarithmic criteria for this class. The criteria describe the behavior of the directional logarithmic derivative outside the zero set and a uniform distribution of zeros in some sense. The criteria are useful for studying analytic solutions of partial differential equations and estimating their growth. We present a scheme of this application.