Generalized Hilbert operator on analytic function spaces

Purpose The main objective of this work is to study the boundedness property of a generalized Hilbert operator Hβ for β ≥ 0 defined by, if f(z)=∑n=0∞anzn be any analytic function on the unit disk D then Hβf(z)=∑n=0∞∑k=0∞Γ(n+β+1)Γ(n+k+1)Γ(n+1)Γ(n+k+β+2)akzn on few analytic function spaces such as Hardy space H1, weighted Hardy spaces HαP and the Bloch space. Design/methodology/approach The authors shall use a few well-known results obtained by using Hardy’s inequality and the Hardy–Littlewood theorem to find the coefficient conditions of an analytic function f on the unit disk D so that f is in Hardy spaces. Also, the authors use many properties of the Gaussian hypergeometric function to prove the main results. Findings We find conditions on the parameter β and establish criteria for the boundedness of Hβ on certain function spaces such as Hardy space H1, weighted Hardy spaces HαP and the Bloch space. Originality/value In particular, β = 0 gives the classical Hilbert operator H. So our results extend some results of Lanucha et al. [10]. The authors use of the Gaussian hypergeometric function in the proof is a different approach.