Boundedness of the L-Index in the Direction of Composition of Slice Entire Functions and Slice Holomorphic Functions in the Unit Ball

We study a composition F ( z ) := f Φ z , . . . , Φ z ⏟ m times : C n → C , n ≥ 1, m ≥ 1, where Φ : B n → C is a slice holomorphic function in the unit ball and f : C m → C is a slice holomorphic function in the whole m -dimensional complex space C m , i.e., a slice function g z ( τ ) = f ( z + b τ ) is an entire function of τ ∈ C for any fixed z ∈ C m and a given direction b ∈ C m . The slice holomorphy in a unit ball B n means that, for a fixed direction b ∈ C n \ { 0 } and any point z 0 ∈ B n of the unit ball, the corresponding slice function is holomorphic on the restriction of the original function to the slice { z 0 + t b : t ∈ C } ∩ B n . An additional assumption about the equicontinuity of these functions allows us to construct an analog of the theory of entire functions with bounded index. The corresponding results of this analog can be applied to the investigation of the properties of slice-holomorphic solutions of directional differential equations that describe local behaviors and the distribution of values. We establish conditions sufficient for the boundedness of the L -index in the direction b for the function F ( z ) . Some of the obtained results are also new in the one-dimensional case, i.e., for n = 1 and m = 1, where the ball is reduced to a unit circle. The indicated conditions are obtained by using two different approaches from the theory of functions of bounded index: an analog of Hayman’s theorem and an analog of the logarithmic criterion. We also present examples of functions whose composition satisfies all conditions of only one of the obtained theorems.