Boundedness of generalized Cesaro averaging operators on certain function spaces

We define a two-parameter family of Cesaro averaging operators P-b,P-c, P-b,P-c f(z) = Gamma(b + 1)/Gamma(c)Gamma(b + 1 - c) integral(1)(0) t(c-1) (1 - t)(b-c) (1 - tz)F(1, b + 1; c; tz) f(tz) dt, where Re (b + 1) > Re c > 0, f(z) = Sigma(infinity)(n=0) a(n)z(n) is analytic on the unit disc Delta, and F (a, b; c; z) is the classical hypergeometric function. In the present article the boundedness of P-b,P-c, Re (b + 1) > Re c > 0, on various function spaces such as Hard,, BMOA and a-Bloch spaces is proved. In the special case b = 1 + alpha and c = 1, P-b,P-c becomes the alpha-Cesdro operator l(a), Re alpha > - 1. Thus, our results connect the special functions in a natural way and extend and improve several well-known results of Hardy-Littlewood, Miao, Stempak and Xiao. (c) 2004 Elsevier B.V. All rights reserved.