### Some weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball

#### Abstract

We partially reinforce some criteria of $L$-index boundedness in direction for functions analytic in the unit ball. These results describe local behavior of directional derivatives on the circle, estimates of maximum modulus, minimum modulus of analytic function, distribution of its zeros and modulus of directional logarithmic derivative of analytic function outside some exceptional set. Replacement of universal quantifier on existential quantifier gives new weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball. The results are also new for analytic functions in the unit disc. The logarithmic criterion has applications in analytic theory of differential equations. This is convenient to investigate index boundedness for entire solutions of linear differential equations. It is also apllicable to infinite products.

Auxiliary class of positive continuous functions in the unit ball (so-denoted $Q_{\mathbf{b}}(\mathbb{B}^n)$) is also considered. There are proved some characterizing properties of these functions. The properties describe local behavior of these functions in the polydisc neighborhood of every point from the unit ball.

Auxiliary class of positive continuous functions in the unit ball (so-denoted $Q_{\mathbf{b}}(\mathbb{B}^n)$) is also considered. There are proved some characterizing properties of these functions. The properties describe local behavior of these functions in the polydisc neighborhood of every point from the unit ball.

#### Keywords

bounded $L$-index in direction, analytic function, unit ball, maximum modulus, directional derivative, distribution of zero

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