Circular Symmetrization, Subordination and Arclength problems on Convex Functions

We study the class $\mathcal{C}(\mathrm{\Omega })$ of univalent analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ of the form $f(z)=z+{\sum }_{n=2}^{\mathrm{\infty }}{a}_n{z}^n$ satisfying $$1+\frac{zf\mathrm{"}(z)}{f^{\mathrm{\prime }}(z)}\in \mathrm{\Omega },z\in \mathbb{D},$$ where $\mathrm{\Omega }$ will be a proper subdomain of $\mathbb{C}$ which is starlike with respect to $1(\in \mathrm{\Omega })$. Let ${\varphi }_{\mathrm{\Omega }}$ be the unique conformal mapping of $\mathbb{D}$ onto $\mathrm{\Omega }$ with ${\varphi }_{\mathrm{\Omega }}(0)=1$ and ${\varphi }_{\mathrm{\Omega }}^{\mathrm{\prime }}(0)>0$ and $ k_\Omega (z) = \int_0^z \exp \left(\int_0^t \zeta^{-1} (\phi_\Omega (\zeta) -1) , d \zeta \right) , dt$\mathrm{.}Let$L_r(f)$denotethearclengthoftheimageofthecircle${z \in $\mathbb{C}$ : , |z|=r}$,$r\in (0,1)$. The first result in this paper is an inequality $L_r(f)\le L_r(k_{\mathrm{\Omega }})$ for $f\in \mathcal{C}(\mathrm{\Omega })$, 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 ${\mathcal C}(\Omega)$.