International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 871845, 29 pages
Research Article

Recent Developments of Hilbert-Type Discrete and Integral Inequalities with Applications

1Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539, USA
2Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, China

Received 7 March 2012; Revised 12 June 2012; Accepted 27 June 2012

Academic Editor: Tibor K. Pogány

Copyright © 2012 Lokenath Debnath and Bicheng Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper deals with recent developments of Hilbert-type discrete and integral inequalities by introducing kernels, weight functions, and multiparameters. Included are numerous generalizations, extensions, and refinements of Hilbert-type inequalities involving many special functions such as beta, gamma, logarithm, trigonometric, hyper-bolic, Bernoulli's functions and Bernoulli's numbers, Euler's constant, zeta function, and hypergeometric functions with many applications. Special attention is given to many equivalent inequalities and to conditions under which the constant factors involved in inequalities are the best possible. Many particular cases of Hilbert-type inequalities are presented with numerous applications. A large number of major books and recent research papers published during 2009–2012 are included to stimulate new interest in future study and research.