ON A CLASS OF FUNCTIONS UNIFYING THE CLASSES OF PAATERO, ROBERTSON AND OTHERS

%(a 6,b,c) of analytic functions which unifies aABSTRACT. We study a class Mk number of classes studied previously by Paatero, Robertson, Pinchuk, Moulis, Mocanu and others. Thus our class includes convex and starlike functions of order 6, spirallike functions of order 6 and functions for which zf ’ is spirallike of order, functions of boundary rotation utmost k, a-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 M (a,6,b,c) which unifies some earlier results concerning the radii of convexity of functions in the class V() of Moulis and those concerning the radii of starlikeness of functions in the classes Uk of Pinchuk and U2(8) of(0)Robertson etc. By applying an estimate of Moulis concerning functions in Vk we obtain an inequality in the class Mk (a,8,b,c) which will contain an estimate for the Schwarzian deriative of functions in the class Vk(8) and in particular the estimate of Moulis for the Schwarzian of functions in V(0)