### Analytic vector-functions in the unit ball having bounded $\mathbf{L}$-index in joint variables

#### Abstract

In this paper, we consider a class of vector-functions which are analytic in the unit ball. For this class of functionsthere is introduced a concept of boundedness of $\mathbf{L}$-index in joint variables where$\mathbf{L}=(l_1,l_2): \mathbb{B}^2\to\mathbb{R}^2_+$ is a positive continuous vector-function,$\mathbb{B}^2=\{z\in\mathbb{C}^2: |z|=\sqrt{|z_1|^2+|z_2|^2}\le 1\}.$We present necessary and sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables.They describe the local behavior of the maximum modulus of every component of the vector-function or its partial derivatives.

#### Keywords

bounded index, bounded $\mathbf{L}$-index in joint variables, analytic function, unit ball, local behavior, maximum modulus, $\sup$-norm, verctorvalued function

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