Circular Symmetrization, Subordination and Arclength problems on Convex Functions

Abstract

We study the class $\mathcal{C}(\Omega )$ of univalent analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ of the form $f(z)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$ satisfying \[ 1+\frac{zf"(z)}{f'(z)} \in Ω, \quad z\in \mathbb{D}, \] where $\Omega$ will be a proper subdomain of $\mathbb{C}$ which is starlike with respect to $1(\in \Omega )$. Let ${\varphi }_{\Omega }$ be the unique conformal mapping of $\mathbb{D}$ onto $\Omega$ with ${\varphi }_{\Omega }(0)=1$ and ${\varphi }_{\Omega }^{\prime }(0)>0$ and $k_{\Omega }(z)={\int }_0^z\exp \left({\int }_0^t{\zeta }^{-1}({\varphi }_{\Omega }(\zeta )-1)d\zeta \right)dt$. Let $L_r(f)$ denote the arclength of the image of the circle $\{z\in \mathbb{C}:|z|=r\}$, $r\in (0,1)$. The first result in this paper is an inequality $L_r(f)\le L_r(k_{\Omega })$ for $f\in \mathcal{C}(\Omega )$, which solves the general extremal problem ${\max }_{f\in \mathcal{C}(\Omega )}{L}_r(f)$, 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 $\mathcal{C}(\Omega )$.

This is appear in Mathematische Nachrichten