Generalized Hilbert-Type Operator on Hardy Space

S. Naik, P. K. Nath

Abstract


If \(f\) be an analytic function on the unit disc \(\mathbb{D}\) with Taylor series expansion \(\displaystyle f(z) = \sum_{n=0}^\infty a_nz^n\), we consider the generalized Hilbert-type operator defined by \(\displaystyle\mathcal{H}_{a,b}(f)(z)=\sum_{n=0}^\infty\left(\sum_{k=0}^\infty \frac{\Gamma(n+a+1)\Gamma(n+k+1)}{\Gamma(n+1)\Gamma(n+k+b+2)}a_k\right)z^n\) where \(\Gamma\) 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.

Keywords


Generalized Hilbert-type operator; Hardy Spaces

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References


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