Generalized Hilbert-Type Operator on Hardy Space - Naik, S.; Nath, P. K. (Communications in Mathematics and Applications, 2015)
If \(f\) be an analytic function on the unit disc \(\mathbb{D}\) with Taylor series expansion \(\displaystyle f(z) = \sum_{n=0}^∞ a_nz^n\), we consider the generalized Hilbert-type operator defined by \(\displaystyle\mathcal{H}_{a,b}(f)(z)=\sum_{n=0}^∞\left(\sum_{k=0}^∞ \frac{Γ(n+a+1)Γ(n+k+1)}{Γ(n+1)Γ(n+k+b+2)}a_k\right)z^n\) where \(Γ\) denotes the Gamma function and \(a, b \in\mathbb{C}\). We find an upper bound for the norm of the generalized Hilbert-type operator on Hardy space.