Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables

Abstract

Abstract In this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-valued functions analytic in the unit ball B 2 = { z ∈ C 2 : | z | = | z 1 | 2 + | z 2 | 2 < 1 } , $\begin{array}{} \mathbb{B}^2\! = \!\{z\!\in\!\mathbb{C}^2: |z|\! = \!\small\sqrt{|z_1|^2+|z_2|^2}\! \lt \! 1\}, \end{array} $ where L = (l 1, l 2): 𝔹2 → R + 2 $\begin{array}{} \mathbb{R}^2_+ \end{array} $ is a positive continuous vector-valued function. Particularly, we deduce analog of Hayman’s theorem for this class of functions. The theorem shows that in the definition of boundedness of L-index in joint variables for vector-valued functions we can replace estimate of norms of all partial derivatives by the estimate of norm of (p + 1)-th order partial derivative. This form of criteria could be convenient to investigate analytic vector-valued solutions of system of partial differential equations because it allow to estimate higher-order partial derivatives by partial derivatives of lesser order. Also, we obtain sufficient conditions for index boundedness in terms of estimate of modulus of logarithmic derivative in each variable for every component of vector-valued function outside some exceptional set by the vector-valued function L(z).