Local behavior of slice holomorphic functions in the unit ball and boundedness of L-index in direction

Abstract

Let b∈ℂn \{0} be a fixed direction and L : 𝔹n→ℝ+ be a positive continuous function such that L(z)>β|b|1−|z| , where β > 1 is some constant. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice {z0+tb : t∈ℂ} with the unit ball 𝔹n={z∈ℂn:|z|:=|z|12+…+|zn|2<1} for any z0∈ 𝔹n. For functions from this class we prove some criteria of boundedness of L-index in the direction describing local behavior of maximum modulus, minimum modulus of the slice holomorphic function and providing estimates of logarithmic derivative and distribution of zeros. Moreover, we obtain an analog of logarithmic criterion. Note that the hypothesis on holomorphy in one direction together with the hypothesis on joint continuity do not imply holomorphy in whole n-dimensional unit ball.