## Rocky Mountain Journal of Mathematics

- Rocky Mountain J. Math.
- Volume 49, Number 4 (2019), 1063-1092.

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

Andriy Bandura and Oleh Skaskiv

#### 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.

#### Article information

**Source**

Rocky Mountain J. Math., Volume 49, Number 4 (2019), 1063-1092.

**Dates**

First available in Project Euclid: 29 August 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.rmjm/1567044028

**Digital Object Identifier**

doi:10.1216/RMJ-2019-49-4-1063

**Mathematical Reviews number (MathSciNet)**

MR3998910

**Zentralblatt MATH identifier**

07104706

**Subjects**

Primary: 32A10: Holomorphic functions

Secondary: 32A17: Special families of functions 35B08: Entire solutions

**Keywords**

Analytic function unit ball bounded $L$-index in direction growth estimates partial differential equation several complex variables

#### Citation

Bandura, Andriy; Skaskiv, Oleh. Functions analytic in the unit ball having bounded $L$-index in a direction. Rocky Mountain J. Math. 49 (2019), no. 4, 1063--1092. doi:10.1216/RMJ-2019-49-4-1063. https://projecteuclid.org/euclid.rmjm/1567044028