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 \[ 1+\frac{zf"(z)}{f'(z)} \in \Omega, \quad z\in \mathbb{D}, \] 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 ${\mathcal C}(\Omega)$.