Boundedness of generalized Cesلro averaging operators on certain function spaces

We define a two-parameter family of Cesáro averaging operators P b , c , P b , c f ( z ) = Γ ( b + 1 ) Γ ( c ) Γ ( b + 1 - c ) ∫ 0 1 t c - 1 ( 1 - t ) b - c ( 1 - tz ) F ( 1 , b + 1 ; c ; tz ) f ( tz ) d t , where Re ( b + 1 ) > Re c > 0 , f ( z ) = ∑ n = 0 ∞ a n z n 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 P b , c , Re ( b + 1 ) > Re c > 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 , P b , c becomes the α -Cesáro operator C α , Re α > - 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.