TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 4, pp. 1271-1282, August 2010 This paper is available online at http://www.tjm.nsysu.edu.tw/ ON NEW HILBERT-PACHPATTE TYPE INTEGRAL INEQUALITIES C.-J. Zhao 1∗ and W.-S. Cheung 2 Abstract. Inverses of some new inequalities similar to Hilbert’s inequality are established. Our results provide new estimates on these types of inequalities. 1. INTRODUCTION In recent years several authors [1-9] have given considerable attention to Hilbert’s inequalities and Hilbert’s type inequalities and their various generalizations. In par- ticular, in 1988, B. G. Pachpatte [1] proved some new integral inequalities similar to Hilbert’s inequality [10, p. 226] , The main purpose of this paper is to establish their inverses. 2. MAIN RESULTS In [1], Pachpatte established the following Hilbert type integral inequality. Theorem A Let h ≥ 1,l ≥ 1 and let f (σ) ≥ 0,g (τ ) ≥ for σ ∈ (0,x),τ ∈ (0,y), where x and y are positive real numbers and define F (s)=  s 0 f (σ)dσ and G(t)=  t 0 g (τ )dτ , for s ∈ (0,x),t ∈ (0,y). Then  x 0  y 0 F h (s)G l (t) s + t dsdt ≤ 1 2 hl (xy) 1/2  x 0  x - s  F h-1 (s)f (s)  2 ds  1/2 (1) ×  y 0  y - t  G l-1 (t)g (t)  2 dt  1/2 . Received May 10, 2008, accepted August 18, 2008. Communicated by Sen-Yen Shaw. 2000 Mathematics Subject Classification: 26D15. Key words and phrases: Hilbert’s inequality, H¨ older integral inequality, Jensen integral inequality. 1 Research is supported by the National Natural Science Foundation of China (Grant No. 10971205). 2 Research is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project No. HKU7016/07P). *Correspondence author. 1271