Circular symmetrization, subordination and arclength problems on convex functions

Article Type

Research Article

Publication Title

Mathematische Nachrichten

Abstract

We study the class C(Ω) of univalent analytic functions f in the unit disk D=z∈C:|z|<1 of the form f(z)=z+∑n=2∞anzn satisfying 1+zf''(z)/f'(z)∈Ω, z∈D, where Ω will be a proper subdomain of C which is starlike with respect to 1(∈Ω). Let φΩ be the unique conformal mapping of D onto Ω with φΩ(0)=1 and φΩ'(0)>0 and kΩ(z)=∫0zexp(∫0tζ-1(φΩ(ζ)-1)dζ)dt. Let Lr(f) denote the arclength of the image of the circle z∈C:|z|=r, r∈(0,1). The first result in this paper is an inequality Lr(f)≤Lr(kΩ;) for f ∈C(Ω), which solves the general extremal problem maxf∈C(Ω)Lr(f), and contains many other well-known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class C(Ω).

First Page

1044

Last Page

1051

DOI

10.1002/mana.201500027

Publication Date

6-1-2016

Comments

Open Access; Green Open Access

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