Koebe’s and Bieberbach’s Inequalities in the Banach Algebra of Continuous Functions

Throughout the paper, C T. denotes the Banach algebra, with sup norm, of continuous complex-valued functions defined on a compact metric space T. By B f; r. we denote an open ball in C T. centered at fgC T. of radius r)0, and by U a; r. the open ball in the complex plane C at agC of radius r)0 a neighborhood of a.. In compliance with usual notations concerning uni¨alent functions, by S we denote the class of functions f z.szq `ps1ap zp uni¨alent in the unit disk EsU 0; 1.. By means of fractional deri¨ati¨e, interesting generalizations of certain inequalities are obtained. The object of the present article is to generalize the class S in the algebra C T. and to prove a generalization of Koebe’s and Bieberbach’s inequalities: If fgS and zgE, then 1y