Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 918534, 12 pages doi:10.1155/2008/918534 Research Article The Generalizations of Hilbert’s Inequality Liubin Hua, 1 Huazhong Ke, 2 and Yongjin Li 2 1 Guangzhou Sontan Polytechnic College, Guangzhou 511370, China 2 Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China Correspondence should be addressed to Yongjin Li, stslyj@mail.sysu.edu.cn Received 24 June 2008; Accepted 2 October 2008 Recommended by Feng Qi By introducing some parameters and estimating the weight functions, we establish a new Hilbert- type inequalities with best constant factors. The equivalent inequalities are also considered. Copyright q 2008 Liubin Hua et al. 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. 1. Introduction If f, g are real functions such that 0 <  ∞ 0 f 2 xdx < ∞ and 0 <  ∞ 0 g 2 xdx < ∞, then we have see 1  ∞ 0  ∞ 0 f xg y x  y dx dy < π  ∞ 0 f 2 xdx  ∞ 0 g 2 xdx  1/2 , 1.1 where the constant factor π is the best possible. Inequality 1.1 is the well-known Hilbert’s inequality. Inequality 1.1 had been generalized by Hardy-Riesz see 2 in 1925 as if f, g are real functions such that 0 <  ∞ 0 f p xdx < ∞ and 0 <  ∞ 0 g q xdx < ∞, then  ∞ 0  ∞ 0 f xg y x  y dx dy < π sinπ/p  ∞ 0 f p xdx  1/p  ∞ 0 g q xdx  1/q , 1.2 where the constant factor c  π/ sinπ/p is the best possible. When p  q  2, 1.2 reduces to 1.1. Inequality 1.2 is named after Hardy-Hilbert’s integral inequality, which is important in the analysis and its applications see 3, it has been studied and generalized in many directions by a number of mathematicians see 4–8.