Circular Symmetrization, Subordination and Arclength problems on Convex Functions

We study the class ${\mathcal C}(Ω)$ 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 $Ω$ will be a proper subdomain of ${\mathbb C}$ which is starlike with respect to $1 (\in Ω)$. Let $ϕ_Ω$ be the unique conformal mapping of ${\mathbb D}$ onto $Ω$ with $ϕ_Ω(0)=1$ and $ϕ_Ω'(0) > 0$ and $k_Ω(z) = \int_0^z \exp \left(\int_0^t ζ^{-1} (ϕ_Ω(ζ) -1) \, d ζ\right) \, dt$. 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 $L_r(f) \leq L_r(k_Ω)$ for $f \in \mathcal{C} (Ω)$, which solves the general extremal problem $\max_{f \in {\mathcal C}(Ω)} L_r(f)$, 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 ${\mathcal C}(Ω)$.

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