Two-point distortion theorems and the Schwarzian derivatives of meromorphic functions (Q1659303)

In the paper, two-point distortion theorems for meromorphic functions \(f\) in the unit disk \(U=\{z\in\mathbb C:|z|<1\}\), involving \(f'\) and the Schwarzian derivative \(S_f\), are obtained under weaker conditions than the univalence property of \(f\). Given \(w_1,w_2\in\mathbb C\), let \(\Gamma(w_1,w_2)\) denote the collection of all circles passing through \(w_1\) and \(w_2\), and let \(\Delta(w_1,w_2)\) denote the family of Jordan curves given by \[ \left|\left(\frac{2w-w_1-w_2}{w_2-w_1}\right)^2-1-it\right|=|t|,\;\;\;t\in\mathbb R,\;\;\;t\neq0, \] that emerge when \(t\) runs over \(\mathbb R\setminus\{0\}\). For a meromorphic function \(f\) in \(U\), denote by \(\mathcal R(f)\) the Riemann surface onto which \(f\) maps \(U\). A set \(\Lambda\subset\mathcal R(f)\) is one-sheeted if different points of \(\Lambda\) have different projections. The author proves the following statements. Theorem 1. Let \(f:U\to\mathbb R(f)\) be meromorphic, and let \(z_1,z_2\in U\) be such that \(f'(z_k)\neq0\), \(k=1,2\), and \(\infty\neq w_1=\text{pr}\,f(z_1)\neq\text{pr}\,f(z_2)=w_2\neq\infty\). Suppose that \(\gamma_1\cup\gamma_2\) is one-sheeted for any pair of non-intersecting Jordan curves \(\gamma_1,\gamma_2\in\mathcal R(f)\) not passing through the ramification points of \(\mathcal R(f)\), lying over the same circle from \(\Gamma(w_1,w_2)\) and such that \(f(z_k)\in\gamma_k\), \(k=1,2\). Then, for the hyperbolic distance \(d(z_1,z_2)\), the inequality \[ (1-|z_1|^2)(1-|z_2|^2)|f'(z_1)f'(z_2)|\tanh^2(d(z_1,z_2))\leq|w_1-w_2|^2 \] holds and is sharp. Corollary 1. Let \(f:U\to\overline{\mathbb C}\) be meromorphic, and let \(z_1,z_2\in U\) be distinct from the poles of \(f\) and such that \(w_1=f(z_1)\neq f(z_2)=w_2\). Suppose that every arc of the circle joining \(f(z_1)\) and \(f(z_2)\) and belonging to \(f(U)\) is univalently covered by \(f\). Then the inequality of Theorem 1 holds and is sharp. Theorem 2. Let \(f:U\to\mathcal R(f)\) be meromorphic, and let \(z_1,z_2\in U\), \(f'(z_k)\neq0\), \(k=1,2\), be such that \(w_k=\text{pr}\,f(z_k)\) satisfy \(w_1\neq w_2\), \(w_k\neq\infty\), \(k=1,2\). Suppose that a Jordan curve on \(\mathcal R(f)\), passing through either of \(f(z_1),f(z_2)\) but not through the ramification points of \(\mathcal R(f)\) and lying over a curve from \(\Delta(z_1,z_2)\), is one-sheeted. Then the sharp estimate \[ \text{Re}\left\{\sum_{k=1}^2\frac{S_f(z_k)(w_2-w_1)^2}{6(f'(z_k))^2}+\frac{2(w_2-w_1)^2}{f'(z_1)f'(z_2)(z_1-z_2)^2}+\right. \] \[ \left. \frac{2|w_2-w_1|^2}{f'(z_1)\overline{f'(z_2)}(1-z_1\overline z_2)^2}\right\}\leq2+\sum_{k=1}^2\frac{|w_2-w_1|^2}{|f'(z_k)|^2(1-|z_k|^2)^2}, \] \[ S_f(z)=\left(\frac{f''(z)}{f'(z)}\right)'-\frac{1}{2}\left(\frac{f''(z)}{f'(z)}\right)^2, \] holds. Equality is attained, for example, for \[ f(z)=\frac{z(1+\lambda^2)-i\lambda(1+z^2)}{\lambda(1+z^2)-iz(1+\lambda^2)} \] and \(z_1=-\lambda\), \(z_2=\lambda\), \(0<\lambda<1\). Each extremal function \(f\) maps \(U\) onto the \(w\)-plane slit along an arc on \(\partial U\) with \(f(\pm\lambda)=\pm1\).