^{1}Institute of Mathematics, University of Zurich, CH-8057, Zurich, Switzerland \ & Institute for Advanced Study, Program in Interdisciplinary Studies, 1 Einstein Dr, Princeton, NJ 08540, USA

^{2}Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P. R. China

Abstract

By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the reverses and some particular cases are also considered.

f $p>1,frac{1}{p}+frac{1}{q}=1,f(x),g(y)geq 0,fin L^{p}(mathbf{R}% _{+}),gin L^{q}(mathbf{R}_{+}),$ $$||f||_{p}=left(int_{0}^{infty }f^{p}(x)dxright)^{frac{1}{p}}>0,$$ and $||g||_{q}>0,$ then we have the following Hardy-Hilbert's integral inequality ...

References

G.H. Hardy, J.E. Littlewood and G. P$acute{o}$lya: {it Inequalities}, Cambridge University Press, Cambridge, 1934.