The generalized Roper–Suffridge extension operator

Abstract

Let f(z) be a normalized convex (starlike) function on the unit disc D. Let , where z=(z₁,z₂,…,z_n), z₁∈D, $\text{(z}{}_2\text{,…,z}{}_{\mathrm{n}}\text{)∈}\text{C}{}^{\mathrm{n-1}}$, p_i⩾1, i=2,…,n, are real numbers. In this note, we prove that Φ(f)(z)=(f(z₁),f′(z₁)^1/p₂z₂,…,f′(z₁)^1/p_nz_n) is a normalized convex (starlike) mapping on $\text{Ω}$, where we choose the power function such that (f′(z₁))^1/p_i|_z1=0=1, i=2,…,n. Some other related results are proved.