Schwarzian derivative for convex mappings of order \(\alpha\) (Q2684879)

The authors obtain sharp bounds for the Schwarzian derivative in a subclass of convex mappings. For holomorphic functions \(f\), \(f(0)=0\), \(f'(0)=1\), in the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\), let \(Sf\) be the Schwarzian derivative \(Sf(z)=(f''(z)/f'(z))'-1/2(f''(z)/f'(z))^2\). Convex functions \(f\) of order \(\alpha\), \(0\leq\alpha<1\), are defined by the condition \(\mathrm{Re}(1+zf''(z)/f'(z))>\alpha\) in \(\mathbb D\). The class of such functions is denoted by \(C_{\alpha}\). The authors give two inequalities which are equivalent to the condition \(f\in C_{\alpha}\). Also, \(|f(z)|\) and \(|f'(z)|\) are estimated for \(f\in C_{\alpha}\). We say that \(f\in C_{\alpha}^0\) if \(f\in C_{\alpha}\) and \(f''(0)=0\). The following theorem presents sharp bounds of \(|Sf(z)|\) in the class \(C_{\alpha}^0\). Theorem 5. If \(f\in C_{\alpha}^0\), \(0\leq\alpha<1\), then \[ (1-|z|^2)^2|Sf(z)|\leq2(1-\alpha^2),\;\;\;z\in\mathbb D. \] The inequality is sharp Also, the authors estimate \((1-|z|^2)^2|Sf(z)|\) for \(f\in C_{\alpha}\) and the given value \(p=|f''(0)|/2(1-\alpha)\).