Two points distortion estimates for convex univalent functions

Abstract

We study the class $\mathcal{CV}(\Omega )$ of 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 $\Omega$ is a convex and proper subdomain of ${\mathbb C}$ with $1 \in \Omega$. 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 $z_0, z_1 \in \mathbb{D}$ with $z_0 \not= z_1$. As the first result in this paper we show that the region of variability $\{ \log f'(z_1) - \log f'(z_0) : f \in \mathcal{CV}(\Omega ) \}$ coincides with the set $\{ \log k_\Omega ' (z_1 z ) - \log k_\Omega ' (z_0 z ) : |z| \leq 1 \}$. The second result deals with the case when $\Omega$ is the right half plane $\mathbb{H} = \{ w \in \mathbb{C} : \Real w > 0 \}$. In this case $\mathcal{CV}(\Omega)$ is identical with the usual normalized class of convex univalent functions on $\mathbb{D}$. And we derive the sharp upper bound for $| \log f'(z_1) - \log f'(z_0) |$, $f \in \mathcal{CV}(\mathbb{H})$. The third result concerns how far two functions in $\mathcal{CV}( \Omega )$ are from each other. Furthermore we determine all extremal functions explicitly.