Boundedness of generalized Cesáro averaging operators on certain function spaces

We define a two-parameter family of Cesáro averaging operators Pb,c, Pb,cf(z)=Γ(b+1)Γ(c)Γ(b+1-c)∫01tc-1(1-t)b-c(1-tz)F(1,b+1;c;tz)f(tz)dt,where Re(b+1)>Rec>0, f(z)=∑n=0∞anzn is analytic on the unit disc Δ, and F(a,b;c;z) is the classical hypergeometric function. In the present article the boundedness of Pb,c, Re(b+1)>Rec>0, on various function spaces such as Hardy, BMOA and a-Bloch spaces is proved. In the special case b=1+α and c=1, Pb,c becomes the α-Cesáro operator Cα, Reα>-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.