We study a classMkλ(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 orderβ, spirallike functions of orderβand functions for whichzf′is spirallike of orderβ, functions of boundary rotation utmostkπ,α-convex functions etc. An integral representation of Paatero and a variational principle of Robertson for the classVkof 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 classMkλ(α,β,b,c)which unifies some earlier results concerning the radii of convexity of functions in the classVkλ(β)of Moulis and those concerning the radii of starlikeness of functions in the classesUkof Pinchuk andU2(β)of Robertson etc. By applying an estimate of Moulis concerning functions inVkλ(0), we obtain an inequality in the classMkλ(α,β,b,c)which will contain an estimate for the Schwarzian derivative of functions in the classVkλ(β)and in particular the estimate of Moulis for the Schwarzian of functions inVkλ(0).
A Subclass of Analytic Functions Related tok-Uniformly Convex and Starlike Functions