Università degli Studi di Trieste. Dipartimento di Matematica e Informatica
Abstract
For β<1, n = 0, 1, 2, . . ., and −π<α≤π, we let
Mn(α,β) denote the family of functions f(z)=z+…
that are analytic in the unit disk and satisfy there the condition
Re{(Dnf)′+2(n+1)1+eiαz(Dnf)′′}>β,
where Dnf(z) is the Hadamard product or convolution of f with
z/(1−z)n+1. We prove the inclusion relations
Mn+1(α,β)⊂Mn(α,β,
and Mn(α,β)<Mn(π,β),β<1.
Extreme points, as well as integral and convolution characterizations, are found.
This leads to coefficient bounds and other extremal properties.
The special cases M0(α,0)≡Lα,
Mn(π,β)≡Mn(β) have previously
been studied [16], [1]