On a class of functions unifying the classes of Paatero, Robertson and others

We study a class M k λ ( 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 z f ′ is spirallike of order β , functions of boundary rotation utmost k π , α -convex functions etc. An integral representation of Paatero and a variational principle of Robertson for the class V k 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 k λ ( α , β , b , c ) which unifies some earlier results concerning the radii of convexity of functions in the class V k λ ( β ) of Moulis and those concerning the radii of starlikeness of functions in the classes U k of Pinchuk and U 2 ( β ) of Robertson etc. By applying an estimate of Moulis concerning functions in V k λ ( 0 ) , we obtain an inequality in the class M k λ ( α , β , b , c ) which will contain an estimate for the Schwarzian derivative of functions in the class V k λ ( β ) and in particular the estimate of Moulis for the Schwarzian of functions in V k λ ( 0 ) .