Generalized Hilbert-Type Operator on Hardy Space

Authors

  • S. Naik Department of Applied Sciences, Gauhati University, Guwahati 781014, Assam
  • P. K. Nath Department of Applied Sciences, Gauhati University, Guwahati 781014, Assam

DOI:

https://doi.org/10.26713/cma.v6i1.271

Keywords:

Generalized Hilbert-type operator, Hardy Spaces

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.

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References

G. E. Andrews, R Askey and R Roy, Special functions, Cambridge University Press, 1999.

A. Aleman and J.A. Cima, An integral operator on Hp and Hardy's inequality, J. Anal. Math. 85 (2001) 157–176.

K. Avetisyan and S. Stevic, Extended Cesaro operators between different Hardy spaces, Appl. Math. Comput. 207 (2009) 346–350.

D.C. Chang, S. Li and S.Stevic, On some integral operators on the unit polydisc and the unit ball, Taiwan. J. Math. 11 (5) (2007) 1251–1286.

D.C. Chang and S.Stevic, Addendum to the paper ”A note on weighted Bergman spaces and the Cesaro operator”, Nagoya Math. J. 180 (2005) 77–90.

E. Diamantopolous, Hilbert matrix on Bergman spaces, Illinois J. Math. 48 (3) (2004) 1067–1028.

E. Diamantopolous and A. Siskakis, Composition operators and the Hilbert matrix, Studia Math., 140(2), 2000, 191–198.

M. Dostanic, M. Jevtic and D. Vukotic, Norm of the Hilbert matrix on Bergman and

Hardy spaces and a theorem of Nehari type. J. Funct. Ana l. 254, 2008, 2800–2815.

P. L. Duren, Theory of $H^p$ spaces, Academic Press, New York, 1981.

G. Hardy, J. E. Littlewood and G. Poley, Inequalities, 2nd edition, Cambridge University Press, 1988.

S. Li, Generalized Hilbert operator on Dirichlet-type space, Applied Mathematics and Computation, 214, 2009, 304–309.

S. Li and S. Stevic, Generalized Hilbert operator and Fejer-Riesz type inequalities on the polydisc, Acta Math. Sci., 29(B)(1), 2009, 191–200.

J. R . Partington , An introduction to Hankel operators, London Math. Soc. student texts, Cambridge University Press, 13, 1988.

N. M. Temme, Special Functions: An introduction to the classical functions of mathematical physics, John Wiley, New York, 1996.

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Published

08-08-2015
CITATION

How to Cite

Naik, S., & Nath, P. K. (2015). Generalized Hilbert-Type Operator on Hardy Space. Communications in Mathematics and Applications, 6(1), 1–8. https://doi.org/10.26713/cma.v6i1.271

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Section

Research Article