Functions whose derivatives take values in a half-plane

Abstract:

We derive sharp upper and lower bounds for $\left | {zf’(z)/f(z)} \right |$ where $f \in R$, i.e. $f$ analytic in ${\mathbf {D}}$ with $f(0) = 0,f’(0) = 1$ and ${e^{i\alpha }}f’(z){\text { > }}0$ in ${\mathbf {D}}$ for a certain $\alpha = \alpha (f) \in {\mathbf {R}}$. The extremal function is $k(z) = - z - 2\log (1 - z)$. This result improves an earlier one of D. K. Thomas.