Distortion Properties of Holomorphic Functions of Several Complex Variables

Abstract

Let Q(D) be a class of functions q, q(0) = 0, |q(z)| < 1 holomorphic in the Reinhardt domain D ⊂ Cn, a and b — arbitrary fixed numbers satisfying the condition — 1 ≤ b < a ≤ 1. 𝔭(a, b; D) — the class of functions p such that p ϵ 𝔭(a, b; D) iff for some q ϵ Q(D) and every z ϵ D. S*(a, b; D) — the class of functions f such that f ϵ S*(a, g; D) iff Sc(a, b; D) — the class of functions q such that q ϵ Sc(a, b; D) iff , where p e 𝔭(a, b; D) and K is an operator of the form for z=z1,z2,…zn. The author obtains sharp bounds on |p(z)|, f(z)| g(z)| as well as sharp coefficient inequalities for functions in 𝔭(a, b; D), S*(a, b; D) and Sc(a, b; D).