| Abstrakt: |
In this paper, we investigate a composition of entire function of several complex variables and analytic function in the unit ball. We modified early known results with conditions providing equivalence of boundedness of L -index in a direction for such a composition and boundedness of l -index of initial function of one variable, where the continuous function L:B n → R + is constructed by the continuous function l:C m → R +. Taking into account new ideas from recent results on composition of entire functions, we remove a condition that a directional derivative of the inner function Φ in the composition does not equal to zero. Instead of the condition we construct a greater function L (z ) for which F (z ) = f (Φ (z ), …, Φ (z ) (m times) has bounded L -index in a direction, where f:C m → C is an entire function of bounded l -index in the direction (1, …, 1 ), Φ:B n → C is an analytic function in the unit ball. We weaken the condition | ∂ k b Φ (z ) | ≤ K | ∂ b Φ (z ) | k for all z ∈ B n, where K ≥ 1 is a constant, b ∈ C n ∖ { 0 } is a given direction and ∂ b F (z ) := n ∑ j = 1 ∂ F (z ) ∂ z j b j, ∂ k b F (z ) := ∂ b (∂ k − 1 b F (z ) ). It is replaced by the condition | ∂ k b Φ (z ) | ≤ K (l (Φ (z ) ) ) 1 / (N 1 (f, l ) + 1 ) | ∂ b Φ (z ) | k, where N 1 (f, l ) is the l -index of the function f in the direction 1 = (1, …, 1 ). The described result is an improvement of previous one. It is also a new result for the one-dimensional case n = 1, m = 1, i.e. for an analytic function Φ in the unit disc and for an entire function f:C → C of bounded l -index. [ABSTRACT FROM AUTHOR] |
