A half-discrete Hilbert-type inequality in the whole plane related to the Riemann zeta function

Abstract

By the use of Hermite–Hadamard’s inequality and weight functions, a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic cotangent function and multi-parameters is given. The constant factor related to the Riemann zeta function is proved to be the best possible. The equivalent forms, two kinds of particular inequalities, the operator expressions and some equivalent reverses are considered.

Keywords:

• Half-discrete Hilbert-type inequality
• Hermite–Hadamard’s inequality
• weight function
• equivalent form
• Riemann zeta function

AMS Subject Classifications:

• 26D15
• 31A10
• 31B10
• 45P05
• 47G10

Acknowledgements

We are grateful for their help. The authors would like to express their thanks to Ts. Batbold, M. Krnic and J. C. Kuang for reading a preliminary version of the paper and for providing useful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported by the Forschungskredit [grant number FK-15-106] of the University of Zurich. The second author was supported by the National Natural Science Foundation [grant number 61370186], Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education [grant number 2015ARF25].