Circular Symmetrization, Subordination and Arclength problems on Convex Functions

TL;DR

This paper investigates the class of univalent functions constrained by subordination conditions, establishing extremal bounds for arclength and integral means, and unifying various known results within this framework.

Contribution

It introduces a general extremal inequality for arclength in the class ${ m extbf{C}}( ext{Omega})$, encompassing many previous results as special cases.

Findings

Proves an inequality $L_r(f) \,\leq\, L_r(k_\Omega)$ for functions in ${\mathcal C}(\Omega)$.

Solves the extremal problem of maximizing arclength for functions in the class.

Provides results on integral means within the same class.

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 \[ 1+\frac{zf"(z)}{f'(z)} \in \Omega, \quad z\in \mathbb{D}, \] where $\Omega$ will be a proper subdomain of ${\mathbb C}$ which is starlike with respect to $1 (\in \Omega)$. Let $\phi_\Omega$ be the unique conformal mapping of ${\mathbb D}$ onto $\Omega$ with $\phi_\Omega (0)=1$ and $\phi_\Omega '(0) > 0$ and $ k_\Omega (z) = \int_0^z \exp \left(\int_0^t \zeta^{-1} (\phi_\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 $L_r(f) \leq L_r(k_\Omega)$ 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 ${\mathcal C}(\Omega)$.