A Simple Proof of the Lower Bounds of Distortion Theorems on the Roper-Suffridge Operator

In this paper,we prove that the generalized Roper-Suffridge extension operator can be embedded in Loewner chains on the unit ball in Hibert spaces,and obtain the fact that the operator keeps the properties of almost spirallike mapping of type β and other α,almost starlikeness of order α,spirallikeness of type β and starlikeness.

Liczberski-Starkov firstfound a lower bound for ｜｜D（f）｜｜ near the origin, where f（z）=（F（z1）,√F1（z1）z2,…,√F＇（z1）zn）＇is the Roper-Suffridge operator on the unit ball Bn in Cn and F is a normalized convex function on the unit disk. Later, Liczberski-Starkov and Hamada-Kohr proved the lower bound holds on the whole unit ball using a complex computation. Here we provide a rather short and easy proof for the lower bound. Similarly, when F is a normalized starlike function on the unit disk, a lower bound of ｜｜D（f）｜｜ is obtained again.