Majorization-subordination theorems for locally univalent functions. III

Abstract:

A quantitative majorization-subordination result of Goluzin and Tao Shah for univalent functions is generalized to ${\mathfrak {n}_\alpha }$, the linear invariant family of locally univalent functions of finite order $\alpha$. If $f(z)$ is subordinate to $F(z)$ in the open unit disc, $f’(0) \geqslant 0$, and $F(z)$ is in ${\mathfrak {n}_\alpha },1.65 \leqslant \alpha < \infty$, then $f’(z)$ is majorized by $F’(z)$ in $|z| \leqslant (\alpha + 1) - {({\alpha ^2} + 2\alpha )^{1/2}}$. The result is sharp.