S. Bhargava and S. Nanjunda Rao

Copyright © 1988 S. Bhargava and S. Nanjunda Rao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a class Mkλ(α,β,b,c) of analytic functions which unifies a number of classes studied previously by Paatero, Robertson, Pinchuk, Moulis, Mocanu and others. Thus our class includes convex and starlike functions of order β, spirallike functions of order β and functions for which zf′ is spirallike of order β, functions of boundary rotation utmost kπ, α-convex functions etc. An integral representation of Paatero and a variational principle of Robertson for the class Vk of functions of bounded boundary rotation, yield some representation theorems and a variational principle for our class. A consequence of these basic theorems is a theorem for this class Mkλ(α,β,b,c) which unifies some earlier results concerning the radii of convexity of functions in the class Vkλ(β) of Moulis and those concerning the radii of starlikeness of functions in the classes Uk of Pinchuk and U2(β) of Robertson etc. By applying an estimate of Moulis concerning functions in Vkλ(0), we obtain an inequality in the class Mkλ(α,β,b,c) which will contain an estimate for the Schwarzian derivative of functions in the class Vkλ(β) and in particular the estimate of Moulis for the Schwarzian of functions in Vkλ(0).