An estimate of the curvature of the images of circles under maps given by convex univalent functions in a disk

S. M. Yugai

Matematičeskie zametki, Tome 53 (1993) no. 1, pp. 133-137

Résumé

We consider the class $S^0_p$, $p=2,3,\dots$ , of holomorphic functions $f(z)=z+\sum_{n=1}^\infty c ^{(p)} _{np+1} z^{np+1}$ that are univalent in the disk $E=\{z:|z|<1\}$, and that map $E$ onto convex domains that have the property of $p$-tuple symmetry of rotation with respect to the origin. We obtain sharp estimates for the curvature $$ K(w)=\frac1{\rho|f'(z)|}\operatorname{Re}\biggl\{1+\frac{(z-z_0)f''(z)}{f'(z)}\biggr\} $$ of images of the circles $\partial D_\rho=\{z\colon z=r_0+\rho e^{i\varphi},\ 0<r_0<1,\ 0<\rho<1-r_0\}$ at the point $w=f(z)$, $z=r_0+\rho=r$, $0<r<1$.