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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 693248, 6 pages doi:10.1155/2008/693248 Research Article On Inverse Hilbert-Type Inequalities Zhao Changjian 1 and Wing-Sum Cheung 2 1 Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou 310018, China 2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong Correspondence should be addressed to Zhao Changjian, chjzhao@163.com Received 14 November 2007; Revised 1 December 2007; Accepted 4 December 2007 Recommended by Martin J. Bohner This paper deals with new inverse-type Hilbert inequalities. Our results in special cases yield some of the recent results and provide some new estimates on such types of inequalities. Copyright q 2008 Z. Changjian and W S. Cheung. 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 Considerable attention has been given to Hilbert inequalities and Hilbert-type inequalities and their various generalizations by several authors including Handley et al. 1,Minzheand Bicheng 2,Minzhe3,Hu4, Jichang 5, Bicheng 6,andZhao7, 8. In 1998, Pachpatte 9 gave some new integral inequalities similar to Hilbert inequality see 10, page 226.In 2000, Zhao and Debnath 11 established some inverse-type inequalities of the above integral inequalities. This paper deals with some new inverse-type Hilbert inequalities which provide some new estimates on such types of inequalities. 2. Main results Theorem 2.1. Let 0 <p i ≤ 1 i  1, ,n and r ≤ 0.Let{a i,m i } be n positive sequences of real numbers defined for m i  1, 2, ,k i ,wherek i i  1, ,n are natural numbers, define A i,m i   m i s i 1 a i,s i , and define A i,0  0.Thenforp −1  q −1  1, p<0 or 0 <p<1, one has k 1  m 1 1 ··· k n  m n 1  n i1 A p i i,m i  1/n  n i1 m r i  n/pr ≥ n  i1 p i k 1/p i  k i  m i 1  k i − m i  1  a i,m i A p i −1 i,m i  q  1/q . 2.1 2 Journal of Inequalities and Applications Proof. By using the following inequality see 10, page 39: h i a h i −1 i,m i  a i,m i − b i,m i  ≤ a h i i,m i − b h i i,m i ≤ h i b h i −1 i,m i  a i,m i − b i,m i  , 2.2 where a i,m i > 0, b i,m i > 0, and 0 ≤ h i ≤ 1 i  1, 2, ,n,weobtainthat A p i i,m i 1 − A p i i,m i ≥ p i A p i −1 i,m i 1  A i,m i 1 − A i,m i   p i a i,m i 1 A p i −1 i,m i 1 , k i −1  m i 0 A p i i,m i 1 − A p i i,m i  A p i i,k i ≥ k i −1  m i 0 p i a i,m i 1 A p i −1 i,m i 1  p i k i  m i 1 a i,m i A p i −1 i,m i , 2.3 thus A p i i,m i ≥ p i m i  s i 1 a i,s i A p i −1 i,s i . 2.4 From inequality 2.4 and in view of the following mean inequality and inverse H ¨ older’s in- equality 10, page 24,wehave n  i1 m 1/n i ≥  1 n n  i1 m r i  1/r , 2.5  n i1 A p i i,m i  1/n  n i1 m r i  n/pr ≥ n  i1 p i  m i  s i 1  a i,s i A p i−1 i,s i  q  1/q . 2.6 Taking the sum of both sides of 2.6 over m i from 1 to k i 1, 2, ,n first and then using again inverse H ¨ older’s inequality, we obtain that k 1  m 1 1 ··· k n  m n 1  n i1 A p i i,m i  1/n  n i1 m r i  n/pr ≥ n  i1 p i  k i  m i 1  m i  s i 1  a i,s i A p i −1 i,s i  q  1/q  ≥ n  i1 p i k 1/p i  k i  m i 1 m i  s i 1  a i,s i A p i −1 i,s i  q  1/q  n  i1 p i k 1/p i  k i  s i 1  k i − s i  1  a i,s i A p i −1 i,s i  q  1/q  n  i1 p i k 1/p i  k i  m i 1  k i − m i  1  a i,m i A p i −1 i,m i  q  1/q . 2.7 This completes the proof. Remark 2.2. Taking n  2,q −2,r −1to2.1, 2.1 becomes k 1  m 1 1 k 2  m 2 1 A p 1 1,m 1 A p 2 2,m 2  m −1 1  m −1 2  −3 ≥ 8p 1 p 2  k 1 k 2  3/2  k 1  m 1 1  k 1 − m 1  1  a 1,m 1 A p 1 −1 1,m 1  −2  −1/2 ×  k 2  m 2 1  k 2 − m 2  1  a 2,m 2 A p 2 −1 2,m 2  −2  −1/2 . 2.8 Z. Changjian and W S. Cheung 3 This is just an inverse form of the following inequality which was proven by Pachpatte 9: k  m1 r  n1 A p m B q n m  n ≤ 1 2 pqkr 1/2  k  m1 k − m  1  a m A p−1 m  2  1/2  r  n1 r − n  1  b n B q−1 n  2  1/2 . 2.9 Theorem 2.3. Let {a i,m i },A i,m i ,k i ,p,andq be as defined in Theorem 2.1.Let{p i,m i } be n positive sequences for m i  1, 2, ,k i i  1, 2, ,n. Set P i,m i   m i s i 1 p i,s i i  1, 2, ,n.Letφ i i  1, 2, ,n be n real-valued nonnegative, concave, and supermultiplicative functions defined on R   0, ∞.Then, k 1  m 1 1 ··· k n  m n 1  n i1 φ i  A i,m i   1/n  n i1 m r i  n/pr ≥M  k 1 ,k 2 , ,k n  n  i1  k i  m i 1  k i − m i  1   p i,m i φ i  a i,m i p i,m i  q  1/q , 2.10 where M  k 1 ,k 2 , ,k n   n  i1  k i  m i 1  φ i  P i,m i  P i,m i  p  1/p . 2.11 Proof. From the hypotheses and by Jensen’s inequality, the means inequality, and inverse H ¨ older’s inequality, we obtain that n  i1 φ i  A i,m i   n  i1 φ i  P i,m i  m i s i 1 p i,s i  a i,s i /p i,s i   m i s i 1 p i,s i  ≥ n  i1 φ i  P i,m i  φ i   m i s i 1 p i,s i  a i,s i /p i,s i   m i s i 1 p i,s i  ≥ n  i1 φ i  P i,m i  P i,m i m i  s i 1 p i,s i φ i  a i,s i p i,s i  ≥ n  i1 φ i  P i,m i  P i,m i m 1/p i  m i  s i 1  p i,s i φ i  a i,s i p i,s i  q  1/q ≥  1 n n  i1 m r i  n/pr n  i1 φ i P i,m i  P i,m i  m i  s i 1  p i,s i φ i  a i,s i p i,s i  q  1/q . 2.12 Dividing both sides of 2.12 by 1/n  n i1 m r i  n/pr and then taking the sum over m i i  1, 2, ,n from 1 to k i and in view of inverse H ¨ older’s inequality,wehave k 1  m 1 1 ··· k n  m n 1  n i1 φ i  A i,m i   1/n  n i1 m r i  n/pr ≥ n  i1  k i  m i 1 φ i  P i,m i  P i,m i  m i  s i 1  p i,s i φ i  a i,s i p i,s i  q  1/q  ≥ n  i1  k i  m i 1  φ i  P i,m i  P i,m i  p  1/p  k i  m i 1 m i  s i 1  p i,s i φ i  a i,s i p i,s i  q  1/q  M  k 1 ,k 2 , ,k n  n  i1  k i  m i 1 m i  s i 1  p i,s i φ i  a i,s i p i,s i  q  1/q  M  k 1 ,k 2 , ,k n  n  i1  k i  m i 1  k i −m i 1   p i,m i φ i  a i,m i p i,m i  q  1/q . 2.13 The proof is complete. 4 Journal of Inequalities and Applications Remark 2.4. Taking n  2,q −2,r −1to2.10, 2.10 becomes k 1  m 1 1 k 2  m 2 1 φ 1  A 1,m 1  φ 2  A 2,m 2   m −1 1  m −1 2  −3 ≥ M  k 1 ,k 2   k 1  m 1 1  k 1 − m 1  1   p 1,m 1 φ 1  a 1,m 1 p 1,m 1  −2  −1/2 ×  k 2  m 2 1  k 2 − m 2  1   p 2,m 2 φ 2  a 2,m 2 p 2,m 2  −2  −1/2 , 2.14 where M  k 1 ,k 2   8  k 1  m 1 1  φ 1  P 1,m 1  P 1,m 1  2/3  3/2  k 2  m 2 1  φ 2  P 2,m 2  P 2,m 2  2/3  3/2 . 2.15 This is just an inverse of the following inequality which was proven by Pachpatte 9: k  m1 r  n1 φ  A m  ψ  B n  m  n ≤ Mk, r  k  m1 k − m  1  p m φ  a m p m  2  1/2 ×  r  n1 r − n  1  q n ψ  b n q n  2  1/2 , 2.16 where Mk, r 1 2  k  m1  φ  P m  P m  2  1/2  r  n1  ψ  Q n  Q n  2  1/2 . 2.17 Similarly, the following theorem also can be established. Theorem 2.5. Let P i,m i , {a i,m i }, {p i,m i },k i ,p,andq be as in Theorem 2.3 and define A i,m i  1/P i,m i   m i s i 1 p i,s i a i,s i , for m i  1, 2, ,k i .Letφ i i  1, 2, ,n be n real-valued, nonnegative, and concave functions defined on R  .Then, k 1  m 1 1 ··· k n  m n 1  n i1 P i,m i φ i  A i,m i   1/n  n i1 m r i  n/pr ≥ n  i1 k 1/p i  k i  m i 1  k i − m i  1  p i,m i φ i  a i,m i  q  1/q . 2.18 The proof of Theorem 2.5 can be completed by following the same steps as in the proof of Theorem 2.3 with suitable changes. Here, we omit the details. Remark 2.6. Taking n  2,q −2,r −1to2.18, 2.18 becomes k 1  m 1 1 k 2  m 2 1 P 1,m 1 P 2,m 2 φ 1  A 1,m 1  φ 2  A 2,m 2   m −1 1  m −1 2  −3 ≥ 8  k 1 k 2  3/2  k 1  m 1 1  k 1 −m 1 1  p 1,m 1 φ 1  a 1,m 1  −2  −1/2  k 2  m 2 1  k 2 −m 2 1  p 2,m 2 φ 2  a 2,m 2  −2  −1/2 . 2.19 Z. Changjian and W S. Cheung 5 This is just an inverse of the following inequality which was proven by Pachpatte 9: k  m1 r  n1 P m Q n φ  A m  ψ  B n  m  n ≤ 1 2 kr 1/2  k  m1 k − m  1  p m φ  a m  2  1/2  r  n1 r − n  1  q n ψ  b n  2  1/2 . 2.20 Remark 2.7. In view of L’H ˆ opital law, we have the following fact: lim r→0  1 n n  i1 m r i  n/pr  exp  n p lim r→0 ln  1/n  n i1 m r i  r   exp  n p lim r→0  n i1 m r i ln m i  n i1 m r i    m 1 ·m 2 ·····m n  1/p . 2.21 Accordingly, in the special case when n  2, p  0.1, and p i,m i  1, let r → 0, then the inequality 2.18 reduces to the following inequality: k 1  m 1 1 k 2  m 2 1 φ 1  A 1,m 1  φ 2  A 2,m 2   m 1 m 2  −2 ≥  k 1 k 2  −1  k 1  m 1 1  k 1 − m 1 1  φ 1  a 1,m 1  1/2  2  k 2  m 2 1  k 2 −m 2 1  φ 2  a 2,m 2  1/2  2 . 2.22 This is just a discrete form of the following inequality which was proven by Zhao and Debnath 11:  x 0  y 0 φ  Fs  ψ  Gt  st −2 ds dt ≥ xy −1   x 0 x − s  φ  fs  1/2 ds  2   y 0 y − t  φ  gt  1/2 dt  2 . 2.23 Acknowledgments The authors cordially thank the anonymous referee for his/her valuable comments which lead to the improvement of this paper. Research is supported by Zhejiang Provincial Natural Science Foundation of China, Grant no. Y605065, Foundation of the Education Department of Zhejiang Province of China, Grant no. 20050392, partially supported by the Research Grants Council of the Hong Kong SAR, China, Project no. HKU7016/07P. References 1 G. D. Handley, J. J. Koliha, and J. E. Pe ˇ cari ´ c, “New Hilbert-Pachpatte type integral inequalities,” Journal of Mathematical Analysis and Applications, vol. 257, no. 1, pp. 238–250, 2001. 2 G. Minzhe and Y. Bicheng, “On the extended Hilbert’s inequality,” Proceedings of the American Mathe- matical Society, vol. 126, no. 3, pp. 751–759, 1998. 6 Journal of Inequalities and Applications 3 G. Minzhe, “On Hilbert’s inequality and its applications,” Journal of Mathematical Analysis and Applica- tions, vol. 212, no. 1, pp. 316–323, 1997. 4 K. Hu, “On Hilbert inequality and its application,” Advances in Mathematics, vol. 22, no. 2, pp. 160–163, 1993. 5 K. 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Debnath, “Some new inverse type Hilbert integral inequalities,” Journal of Mathemat- ical Analysis and Applications, vol. 262, no. 1, pp. 411–418, 2001. . Corporation Journal of Inequalities and Applications Volume 2008, Article ID 693248, 6 pages doi:10.1155/2008/693248 Research Article On Inverse Hilbert-Type Inequalities Zhao Changjian 1 and Wing-Sum. distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Considerable attention has been given to Hilbert inequalities and Hilbert-type inequalities and. Minzhe, On Hilbert’s inequality and its applications,” Journal of Mathematical Analysis and Applica- tions, vol. 212, no. 1, pp. 316–323, 1997. 4 K. Hu, On Hilbert inequality and its application,”

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