A Class of Nonlinear Averaging Integral Operators

Let H be the class of analytic functions defined in the unit disc U, and let co E denote the convex hull of a set E in C. If K , H, then an operator I :K R H is an averaging operator if I[ f ](0) 5 f (0) and I[ f ](U) , co f (U), for all f [ K. The authors show that the operator Ib,c[ f ](z) ; [cz2c ez 0 f b(t)tc21 dt]1/b is an averaging operator on certain subsets of H. Ó 1996 Academic Press, Inc.