On a Hilbert-Type Integral Inequality in the Whole Plane Related to the Extended Riemann Zeta Function

Abstract

In the present paper, a few equivalent conditions of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The best possible constant factor is related to the extended Riemann zeta function. In the form of applications, a few equivalent conditions of a Hilbert-type integral inequality with the homogeneous kernel in the whole plane are deduced. We also consider the operator expressions and a few particular cases.

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Acknowledgements

M. Th. Rassias: I would like to express my gratitude to the J. S. Latsis Foundation for their financial support provided under the auspices of my current “Latsis Foundation Senior Fellowship” position. B. Yang: This work is supported by the National Natural Science Foundation (Nos. 61370186, 61640222), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25). I feel grateful for this help.

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Correspondence to Michael Th. Rassias.

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Communicated by Daniel Aron Alpay.

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Rassias, M.T., Yang, B. On a Hilbert-Type Integral Inequality in the Whole Plane Related to the Extended Riemann Zeta Function. Complex Anal. Oper. Theory 13, 1765–1782 (2019). https://doi.org/10.1007/s11785-018-0830-5

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Keywords

  • Hilbert-type integral inequality
  • Kernel
  • Weight function
  • Equivalent form
  • Operator
  • Norm

Mathematics Subject Classification

  • 26D15
  • 47A07
  • 65B10