On a class of functions unifying the classes of Paatero, Robertson and others
We study a class 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 , spirallike functions of order and functions for which is spirallike of order , functions of boundary rotation utmost , -convex functions etc. An integral representation of Paatero and a variational principle of Robertson for the class 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 which unifies some earlier results concerning the radii of convexity of functions in the class of Moulis and those concerning the radii of starlikeness of functions in the classes of Pinchuk and of Robertson etc. By applying an estimate of Moulis concerning functions in , we obtain an inequality in the class which will contain an estimate for the Schwarzian derivative of functions in the class and in particular the estimate of Moulis for the Schwarzian of functions in .