A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions (Q1027744)

The authors use Roth's result to obtain a sharp Fekete-Szegő type inequality for \(|b_2+\mu b_1^2|\) in the class of Bloch functions \(F(z)=b_1z+b_2z^2+\dots\), \(|z|<1\), with \(\|F\|=\sup_{|z|<1}(1-|z|^2)|F'(z)|\leq1\) and \(\mu\in\mathbb C\). This is applied to derive a sharp inequality for \(|a_3|\) in the subclass \(\mathcal U(\lambda)\), \(\lambda>0\), of uniformly locally univalent functions \(f(z)=z+a_2z^2+a_3z^3+\dots\), \(|z|<1\), such that \(\sup_{|z|<1}|f''(z)/f'(z)|\leq\lambda\). More generally, \(|a_3-\mu a_2^2|\), \(\mu\in\mathbb C\), is sharply estimated for \(f\in\mathcal U(\lambda)\), \(\lambda>0\).