Generalization of Hilbert-Hardy Integral Inequalities


  • Sobhy a Mahrouf Department of Mathematics, Faculty of Science, Minoufiya University, Shebin El-Koom, Egypt.


Best possible constant, Beta and Gamma functions, Hardy inequality, Hilbert's integral inequality, Weight coefficient.


In this paper, Hilbert's integral inequalities with some parameters are considered, by using new methods in the proof. Several results of Hardy and Yang are special cases of the new given inequality. As an application, we give some applied examples that illustrate our results.


Andrews, L.C. (1985). Special functions of mathematics for

engineers, second ed., McGraw- Hill International Editions.

Bashan, A., Karakoc, S.B.G. & Geyikli, T. (2015).

Approximation of the KdVB equation by thequintic B-spline

differential quadrature method, Kuwait Journal of Science,

(2):67- 92.

Garg, M. & Chanchlani, L. (2013). Solution of nonlinear

q-Shrodinger equation by two dimensional q-differential

transformation method, Kuwait Journal of Science, 40(2):17- 30.

El-Marouf, S.A.A. & AL-Oufi, S.A. (2012). On Opial's

inequality for functions of n-independent variables,

Australian journal of mathematical analysis and applications,

:1 -10.

El-Marouf, S.A.A. (2013). On some generalization of

Hilbert–Hardy type integral inequalities, Indian Journal of

Science and Technology, 6(2):4098–4111.

El-Marouf, S.A.A. (2014). On some generalizations of the

Hilbert–Hardy type discrete

inequalities,Journal of the Egyptian Mathematical Society,


El-Marouf, S.A.A. (2015). Generalization of Hilbert inequality

with some Parameters, Mathematica Slovaca, 65(3):493–510.

Hardy,G.H.,Littlewood, J.E.&Polya, G. (1964). Inequalities,

Cambridge Univ. Press, Cambridge.

Mitrinovic, D. S. (1970). Analytic inequalities, Springer –

Verlag New York .Heidelberg, Berlin.

Mingzhe, G. (1997). Note on Hilbert’s inequality and

its applications. Journal of Mathematical Analysis and

Applications, 212 (1):316- 323.

Salem, S. R. (2006). Some new Hilbert type inequalities ,

KYUNGPOOK Mathematical Journal,4619- 29.

Peachey, T.C. (2003). Some integral inequalities related

to Hilbert’s , Journal of Inequalities in Pure and Applied

Mathematics, 4(1)Art.19:1- 8.

Weijian, J. & Mingzhe, G. (2006). An extended Hardy-

Hilbert Inequality and its applications. Journal of Inequalities

in Pure and Applied Mathematics,7(1)Art.30:1 -9.

Yang, B. (2000). On new generalizations of Hilbert’s

Inequality, Journal of Mathematical Analysis and

Applications, 248:29- 40.

Yang, B. (2004).On the extended Hilbert’s integral inequality.

Journal of Inequalities in Pure and Applied Mathematics,

(4)Art. 96:1 -19.