Composition of entire and analytic functions in the unit ball
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 fC^m→ C is an entire function of bounded l-index in the direction (1,…,1), ΦB^n→ C is an analytic function in the unit ball. We weaken the condition |_b ^kΦ(z)|K|_b Φ(z)|^k for all zB^n, where K≥ 1 is a constant, bC^n{0} is a given direction and _b F(z):=_j=1^n\!F(z)z_j b_j , _b ^k F(z):=_b (_b ^k-1 F(z)). It is replaced by the condition |_b ^kΦ(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. , number=1, journal=Carpathian Mathematical Publications, author=Bandura, A.I. and Skaskiv, O.B. and Tymkiv, I.R., year=2022, month=Jun., pages=95–104