## Rocky Mountain Journal of Mathematics

### 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.

#### Article information

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

Dates
First available in Project Euclid: 29 August 2019

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

#### 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

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