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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 851360, 10 pages doi:10.1155/2009/851360 Research Article Some New Hilbert’s Type Inequalities Chang-Jian Zhao 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 Chang-Jian Zhao, chjzhao@163.com Received 25 December 2008; Accepted 24 April 2009 Recommended by Peter Pang Some new inequalities similar to Hilbert’s type inequality involving series of nonnegative terms are established. Copyright q 2009 C J. Zhao 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 In recent years, several authors 1–10 have given considerable attention to Hilbert’s type inequalities and their various generalizations. In particular, in 1, Pachpatte proved some new inequalities similar to Hilbert’s inequality 11, page 226 involving series of nonnegative terms. The main purpose of this paper is to establish their general forms. 2. Main Results In 1, Pachpatte established the following inequality involving series of nonnegative terms. Theorem A. Let p ≥ 1,q ≥ 1, and let {a m } and {b n } be two nonnegative sequences of real numbers defined for m 1, ,k, and n 1, ,r,wherek,r are natural numbers. Let A m m s1 a s and B n n t1 b t .Then k m1 r n1 A p m B q n m n ≤ C p, q,k, r k m1 k − m 1 a m A p−1 m 2 1/2 × r n1 r − n 1 b n B q−1 n 2 1/2 , 2.1 2 Journal of Inequalities and Applications where C p, q,k, r 1 2 pq kr 1/2 . 2.2 We first establish the following general form of inequality 2.1. Theorem 2.1. Let p ≥ 1,q ≥ 1,t > 0, and 1/α 1/β 1,α > 1.Let{a m 1 , ,m n }, and {b n 1 , ,n n } be positive sequences of real numbers defined for m i 1, 2, ,k i , and n i 1, 2, ,r i ,where k i ,r i i 1, ,n are natural numbers. Let A m 1 , ,m n m 1 s 1 1 ··· m n s n 1 a s 1 , ,s n , and B n 1 , ,n n n 1 t 1 1 ··· n n t n 1 b t 1 , ,t n . Then k 1 m 1 1 ··· k n m n 1 r 1 n 1 1 ··· r n n n 1 αβt 1/β A p m 1 , ,m n B q n 1 , ,n n m 1 ···m n β n 1 ···n n αt ≤ L k 1 , ,k n ,r 1 , ,r n ,p,q,α,β × k 1 m 1 1 ··· k n m n 1 n j1 k j − m j 1 a m 1 , ,m n A p−1 m 1 , ,m n β 1/β × ⎛ ⎝ r 1 n 1 1 ··· r n n n 1 n j1 r j − n j 1 b n 1 , ,n n B q−1 n 1 , ,n n α ⎞ ⎠ 1/α , 2.3 where L k 1 , ,k n ,r 1 , ,r n ,p,q,α,β pq k 1 ···k n 1/α r 1 ···r n 1/β . 2.4 Proof. By using the following inequality see 12: n 1 m 1 1 ··· n n m n 1 z m 1 , ,m n p ≤ p n 1 m 1 1 , , n n m n 1 z m 1 , ,m n m 1 k 1 1 ··· m n k n 1 z k 1 , ,k n p−1 , 2.5 where p ≥ 1 is a constant, and z m 1 , ,m n ≥ 0,m i 1, 2, ,k i , i 1, 2, ,n,weobtain A p m 1 , ,m n ≤ p m 1 s 1 1 ··· m n s n 1 a s 1 , ,s n A p−1 s 1 , ,s n . 2.6 Similarly, we have B q n 1 , ,n n ≤ q n 1 t 1 1 ··· n n t n 1 b t 1 , ,t n B q−1 t 1 , ,t n . 2.7 Journal of Inequalities and Applications 3 From 2.6 and 2.7,usingH ¨ older’s inequality 13 and the elementary inequality: a 1/α b 1/β ≤ a αt 1/β t 1/α b β , 2.8 where 1/α 1/β 1,α>1,b>0,a>0, and t>0, we have A p m i , ,m n B q n i , ,n n ≤ pq m 1 s 1 1 ··· m n s n 1 a s 1 , ,s n A p−1 s 1 , ,s n n 1 t 1 1 ··· n n t n 1 b t 1 , ,t n B q−1 t 1 , ,t n ≤ pq m 1 , ,m n 1/α m 1 s 1 1 ··· m n s n 1 a s 1 , ,s n A p−1 s 1 , ,s n β 1/β × n 1 , ,n n 1/β n 1 t 1 1 ··· n n t n 1 b t 1 , ,t n B q−1 t 1 , ,t n α 1/α ≤ pq m 1 ···m n αt 1/β n 1 ···n n t 1/α β m 1 s 1 1 ··· m n s n 1 a s 1 , ,s n A p−1 s 1 , ,s n β 1/β × n 1 t 1 1 n n t n 1 b t 1 , ,t n B q−1 t 1 , ,t n α 1/α . 2.9 Dividing both sides of 2.9 by m 1 ···m n β n 1 ···n n αt/αβt 1/β , summing up over n i from 1tor i i 1, 2, ,n first, then summing up over m i from 1 to k i i 1, 2, ,n, using again H ¨ older’s inequality, then interchanging the order of summation, we obtain k 1 m 1 1 ··· k n m n 1 r 1 n 1 1 ··· r n n n 1 αβt 1/β A p m 1 , ,m n B q n 1 , ,n n m 1 ···m n β n 1 ···n n αt ≤ pq ⎧ ⎨ ⎩ k 1 m 1 1 ··· k n m n 1 m 1 s 1 1 ··· m n s n 1 a s 1 , ,s n A p−1 s 1 , ,s n β 1/β ⎫ ⎬ ⎭ × ⎧ ⎨ ⎩ r 1 n 1 1 ··· r n n n 1 n 1 t 1 1 ··· n n t n 1 b t 1 , ,t n B q−1 t 1 , ,t n α 1/α ⎫ ⎬ ⎭ ≤ pq k 1 ···k n 1/α k 1 m 1 1 ··· k n m n 1 m 1 s 1 1 ··· m n s n 1 a s 1 , ,s n A p−1 s 1 , ,s n β 1/β × r 1 ···r n 1/β k 1 n 1 1 ··· r n n n 1 n 1 t 1 1 ··· n n t n 1 b t 1 , ,t n B q−1 t 1 , ,t n α 1/α 4 Journal of Inequalities and Applications L k 1 , ,k n ,r 1 , ,r n ,p,q,α,β × k 1 s 1 1 ··· k n s n 1 a s 1 , ,s n A p−1 s 1 , ,s n β k 1 m 1 s 1 ··· k n m n s n 1 1/β × r 1 t 1 1 ··· r n t n 1 b t 1 , ,t n B q−1 t 1 , ,t n α r 1 n 1 t 1 ··· r n n n t n 1 1/α L k 1 , ,k n ,r 1 , ,r n ,p,q,α,β × ⎧ ⎨ ⎩ k 1 s 1 1 ··· k n s n 1 n j1 k j − s j 1 a s 1 , ,s n A p−1 s 1 , ,s n β ⎫ ⎬ ⎭ 1/β × ⎧ ⎨ ⎩ r 1 t 1 1 ··· r n t n 1 n j1 r j − t j 1 b t 1 , ,t n B q−1 t 1 , ,t n α ⎫ ⎬ ⎭ 1/α L k 1 , ,k n ,r 1 , ,r n ,p,q,α,β × ⎧ ⎨ ⎩ k 1 m 1 1 ··· k n m n 1 n j1 k j − m j 1 a m 1 , ,m n A p−1 m 1 , ,m n β ⎫ ⎬ ⎭ 1/β × ⎧ ⎨ ⎩ r 1 n 1 1 ··· r n n n 1 n j1 r j − n j 1 b n 1 , ,n n B q−1 n 1 , ,n n α ⎫ ⎬ ⎭ 1/α . 2.10 This completes the proof. Remark 2.2. Taking α β n j 2, 2.3 becomes k 1 m1 k 2 m 2 1 r 1 n 1 1 r 2 n 2 1 A p m 1 ,m 2 B q n 1 ,n 2 m 1 m 2 t −1/2 n 1 n 2 t 1/2 ≤ 1 2 pq k 1 k 2 r 1 r 2 k 1 m1 k 2 m 2 1 k 1 − m 1 1 k 2 − m 2 1 a m 1 ,m 2 A p−1 m 1 ,m 2 2 1/2 × r 1 n1 r 2 n 2 1 r 1 − n 1 1 r 2 − n 2 1 b n 1 ,n 2 B p−1 n 1 ,n 2 2 1/2 . 2.11 Taking t 1, and changing {a m 1 ,m 2 }, {b n 1 ,n 2 }, {A m 1 ,m 2 }, and {B n 1 ,n 2 } into {a m }, {b n }, {A m }, and {B n }, respectively, and with suitable changes, 2.11 reduces to Pachpatte 1, inequality 1. In 1, Pachpatte also established the following inequality involving series of nonneg- ative terms. Journal of Inequalities and Applications 5 Theorem B. Let {a m }, {b n },A m ,B n be as defined in Theorem A. Let {p m } and {q n } be positive sequences for m 1, ,k,and n 1, ,r, where k, r are natural numbers. Define P m m s1 p s , and Q n n t1 q t .Letφ and ψ be real-valued, nonnegative, convex, submultiplicative functions defined on R 0, ∞. Then k m1 r n1 φ A m ψ B n m n ≤ M k, r k m1 k − m 1 p m φ a m p m 2 1/2 × r n1 r − n 1 q n φ b n q n 2 1/2 , 2.12 where M k, r 1 2 k m1 φ P m P m 2 1/2 r n1 φ Q n Q n 2 1/2 . 2.13 Inequality 2.12 can also be generalized to the following general form. Theorem 2.3. Let {a m 1 , ,m n }, {b n 1 , ,n n }, α, β, t, A m 1 , ,m n , and B n 1 , ,n n be as defined in Theorem 2.1. Let {p m 1 , ,m n } and {q n 1 , ,n n } be positive sequences for m i 1, 2, ,k i , and n i 1, 2, ,r i i 1, 2, ,n. Define P m 1 , ,m n m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n , and Q n 1 , ,n n n 1 t 1 1 ··· n n t n 1 q t 1 , ,t n . Let φ and ψ be real-valued, nonnegative, convex, submultiplicative functions defined on R 0, ∞. Then k 1 m 1 1 ··· k n m n 1 r 1 n 1 1 ··· r n n n 1 αβt 1/β φ A m 1 , ,m n ψ B n 1 , ,n n m 1 ···m n β n 1 ···n n αt ≤ M k 1 , ,k n ,r 1 , ,r n ,α,β × ⎧ ⎨ ⎩ k 1 m 1 1 ··· k n m n 1 n j1 k j − m j 1 p m 1 , ,m n φ a m 1 , ,m n p m 1 , ,m n β ⎫ ⎬ ⎭ 1/β × ⎧ ⎨ ⎩ r 1 n 1 1 ··· r n n n 1 n j1 r j − n j 1 q n 1 , ,n n ψa b n 1 , ,n n q n 1 , ,m n α ⎫ ⎬ ⎭ 1/α , 2.14 where M k 1 , ,k n ,r 1 , ,r n ,α,β k 1 m 1 1 ··· k n m n 1 φ P m 1 , ,m n P m 1 , ,m n α 1/α r 1 n 1 1 ··· r n n n 1 ψQ n 1 , ,n n Q n 1 , ,n n β 1/β . 2.15 6 Journal of Inequalities and Applications Proof. By the hypotheses, Jensen’s inequality, and H ¨ older’s inequality, we obtain φ A m 1 , ,m n φ P m 1 , ,m n m 1 s 1 1 m n s n 1 p s 1 , ,s n a s 1 , ,s n /p s 1 , ,s n m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n ≤ φ P m 1 , ,m n φ m 1 s 1 1 ··· m n s n 1 p s 1 ,··· ,s n a s 1 , ,s n /p s 1 , ,s n m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n ≤ φ P m 1 , ,m n P m 1 , ,m n m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n φ a s 1 , ,s n p s 1 , ,s n ≤ φ P m 1 , ,m n P m 1 , ,m n m 1 ···m n 1/α m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n φ a s 1 , ,s n p s 1 , ,s n β 1/β . 2.16 Similarly, φ B n 1 , ,n n ≤ φ Q n 1 , ,n n Q n 1 , ,n n n 1 ···n n 1/β n 1 n 1 1 ··· n n n n 1 q t 1 , ,t n ψ b t 1 , ,t n q t 1 , ,t n α 1/α . 2.17 By 2.16 and 2.17, and using the elementary inequality: a 1/α b 1/β ≤ a αt 1/β t 1/α b β , 2.18 where 1/α 1/β 1,α>1,b>0,a>0, and t>0, we have φ A m 1 , ,m n φ B n 1 , ,n n ≤ m 1 ···m n αt 1/β n 1 ···n n t 1/α β × φ P m 1 , ,m n P m 1 , ,m n m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n φ a s 1 , ,s n p s 1 , ,s n β 1/β × φ Q n 1 , ,n n Q n 1 , ,n n n 1 t 1 1 n n t n 1 q t 1 , ,t n ψ b t 1 , ,t n q t 1 , ,t n α 1/α . 2.19 Journal of Inequalities and Applications 7 Dividing both sides of 2.19 by m 1 ···m n β n 1 ···n n αt/αβt 1/β , and summing up over n i from 1 to r i i 1, 2, ,n first, then summing up over m i from 1 to k i i 1, 2, ,n,using again inverse H ¨ older’s inequality, and then interchanging the order of summation, we obtain k 1 m 1 1 ··· k n m n 1 r 1 n 1 1 ··· r n n n 1 αβt 1/β φ A m 1 , ,m n ψ B n 1 , ,n n m 1 m n β n 1 n n αt ≤ k 1 m 1 1 k n m n 1 ⎛ ⎝ φ P m 1 , ,m n P m 1 , ,m n m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n φ a s 1 , ,s n p s 1 , ,s n β 1/β ⎞ ⎠ × r 1 n 1 1 ··· r n n n 1 ⎛ ⎝ φ Q n 1 , ,n n Q n 1 , ,n n n 1 n 1 1 ··· n n n n 1 q t 1 , ,t n ψ b t 1 , ,t n q t 1 , ,t n α 1/α ⎞ ⎠ ≤ k 1 m 1 1 ··· k n m n 1 φP m 1 , ,m n P m 1 , ,m n α 1/α × k 1 m 1 1 ··· k n m n 1 m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n φ a s 1 , ,s n p s 1 , ,s n β 1/β × r 1 n 1 1 ··· r n n n 1 ψQ n 1 , ,n n Q n 1 , ,n n β 1/β × r 1 n 1 1 ··· r n n n 1 n 1 t 1 1 ··· n n t n 1 q t 1 , ,t n ψ b t 1 , ,t n q t 1 , ,t n α 1/α M k 1 , ,k n ,r 1 , ,r n ,α,β × ⎧ ⎨ ⎩ k 1 m 1 1 ··· k n m n 1 n j1 k j − m j 1 p m 1 , ,m n φ a m 1 , ,m n p m 1 , ,m n β ⎫ ⎬ ⎭ 1/β × ⎧ ⎨ ⎩ r 1 n 1 1 ··· r n n n 1 n j1 r j − n j 1 q n 1 , ,n n ψ b n 1 , ,n n q n 1 , ,m n α ⎫ ⎬ ⎭ 1/α . 2.20 The proof is complete. 8 Journal of Inequalities and Applications Remark 2.4. Taking α β n j 2, 2.14 becomes k 1 m1 k 2 m 2 1 r 1 n 1 1 r 2 n 2 1 φ A m 1 ,m 2 ψ B n 1 ,n 2 m 1 m 2 t −1/2 n 1 n 2 t 1/2 ≤ M k 1 ,k 2 ,r 1 ,r 2 × k 1 m1 k 2 m 2 1 k 1 − m 1 1 k 2 − m 2 1 p m 1 ,m 2 φ a m 1 , ,m n p m 1 , ,m n 2 1/2 × r 1 n1 r 2 n 2 1 r 1 − n 1 1 r 2 − n 2 1 q n 1 ,n 2 ψ b n 1 , ,n n q n 1 , ,n n 2 1/2 , 2.21 where M k 1 ,k 2 ,r 1 ,r 2 k 1 m 1 1 k 2 m 2 1 φP m 1 ,m 2 P m 1 ,m 2 2 1/2 r 1 n 1 1 r 2 n 2 1 ψQ n 1 ,n 2 Q n 1 ,n 2 2 1/2 . 2.22 Taking t 1, and changing {a m 1 ,m 2 }, {b n 1 ,n 2 }, {A m 1 ,m 2 }, and {B n 1 ,n 2 } into {a m }, {b n }, {A m }, and {B n }, respectively, and with suitable changes, 2.21 reduces to Pachpatte 1, Inequality 7. Theorem 2.5. Let {a m 1 , ,m n }, {b n 1 , ,n n }, {p m 1 , ,m n }, {q n 1 , ,n n }, P m 1 , ,m n , and Q n 1 , ,n n , α,β, t, be as defined in Theorem 2.3. Define A m 1 , ,m n 1 P m 1 , ,m n m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n a s 1 , ,s n , B n1, ,n2 1 Q n 1 , ,n n n 1 t 1 1 ··· n n t n 1 q t 1 , ,t n b t 1 , ,t n , 2.23 for m i 1, 2, ,k i , and n i 1, 2, ,r i i 1, 2, ,n, where k i ,r i i 1, ,n are natural numbers. Let φ and ψ be real-valued, nonnegative, convex functions defined on R 0, ∞. Then k 1 m 1 1 ··· k n m n 1 r 1 n 1 1 ··· r n n n 1 αβt 1/β P m 1 , ,m n Q n 1 , ,n n φ A m 1 , ,m n ψ B n 1 , ,n n m 1 ···m n β n 1 ···n n αt k 1 ···k n 1/α r 1 ···r n 1/β × ⎧ ⎨ ⎩ k 1 m 1 1 ··· k n m n 1 n j1 k j − m j 1 p m 1 , ,m n φ a m 1 , ,m n β ⎫ ⎬ ⎭ 1/β × ⎧ ⎨ ⎩ r 1 n 1 1 ··· r n n n 1 n j1 r j − n j 1 q n 1 , ,n n ψ b n 1 , ,n n α ⎫ ⎬ ⎭ 1/α . 2.24 Journal of Inequalities and Applications 9 Proof. By the hypotheses, Jensen’s inequality, and H ¨ older’s inequality, it is easy to observe that φ A m 1 , ,m n φ 1 P m 1 , ,m n m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n a s 1 , ,s n ≤ 1 P m 1 , ,m n m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n φ a s 1 , ,s n ≤ 1 P m 1 , ,m n m 1 ···m n 1/α m 1 s 1 1 ··· m n s n 1 p s 1 , ,s n φ a s 1 , ,s n β 1/β , 2.25 ψ B n 1 , ,n n ψ 1 Q n 1 , ,n n n 1 t 1 1 ··· n n t n 1 q t 1 , ,t n b t 1 , ,t n ≤ 1 Q n 1 , ,n n n 1 t 1 1 ··· n n t n 1 q t 1 , ,t n ψ b t 1 , ,t n ≤ 1 Q n 1 , ,n n n 1 ···n n 1/β n 1 t 1 1 ··· n n t n 1 q t 1 , ,t n ψ b t 1 , ,t n α 1/α . 2.26 Proceeding now much as in the proof of Theorems 2.1 and 2.3, and with suitable modifica- tions, it is not hard to arrive at the desired inequality. The details are omitted here. Remark 2.6. In the special case where j 1,t 1,α β 2, and n 1, Theorem 2.5 reduces to the following result. Theorem C. Let {a m }, {b n }, {p m }, {q n },P m ,Q n be as defined in Theorem B. Define A m 1/P m m s1 p s a s , and B n 1/Q n n t1 q t b t for m 1, ,k,and n 1, ,r,wherek,r are natural numbers. Let φ and ψ be real-valued, nonnegative, convex functions defined on R 0, ∞. Then k m1 r n1 P m Q n φ A m ψ B n m n ≤ 1 2 kr 1/2 k m1 k − m 1 p m φa m 2 1/2 × r n1 r − n 1q n ψb n 2 1/2 . 2.27 This is the new inequality of Pachpatte in [1, Theorem 4]. Remark 2.7. Taking j 1,t 1,p m 1,q n 1,α β 2, and n 1inTheorem 2.5, and in view of P m m, Q n n, we obtain the following theorem. 10 Journal of Inequalities and Applications Theorem D. Let {a m }, {b n } be as defined in Theorem A. Define A m 1/m m s1 a s , and B n 1/n n t1 b t , for m 1, ,k, and n 1, ,r,wherek, r are natural numbers. Let φ and ψ be real-valued, nonnegative, convex f unctions defined on R 0, ∞. Then k m1 r n1 mn m n φ A m ψ B n ≤ 1 2 kr 1/2 k m1 k − m 1 φa m 2 1/2 × r n1 r − n 1 ψ b n 2 1/2 . 2.28 This is the new inequality of Pachpatte in [1, Theorem 3]. Acknowledgments Research is supported by Zhejiang Provincial Natural Science Foundation of ChinaY605065, Foundation of the Education Department of Zhejiang Province of China 20050392. Research is partially supported by the Research Grants Council of the Hong Kong SAR, China Project no. HKU7016/07P and a HKU Seed Grant forBasic Research. References 1 B. G. Pachpatte, “On some new inequalities similar to Hilbert’s inequality,” Journal of Mathematical Analysis and Applications, vol. 226, no. 1, pp. 166–179, 1998. 2 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. 3 M. Gao and B. Yang, “On the extended Hilbert’s inequality,” Proceedings of the American Mathematical Society, vol. 126, no. 3, pp. 751–759, 1998. 4 K. 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Pe ˇ cari ´ c, “A Hilbert type inequality,” Tamkang Journal of Mathematics, vol. 31, no. 4, pp. 311–315, 2000. 11 G. H. Hardy, J. E. Littlewood, and G. P ´ olya, Inequalities, Cambridge University Press, Cambridge, UK, 1934. 12 J. N ´ emeth, “Generalizations of the Hardy-Littlewood inequality,” Acta Scientiarum Mathematicarum, vol. 32, pp. 295–299, 1971. 13 E. F. Beckenbach and R. Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 30, Springer, Berlin, Germany, 1961. . Corporation Journal of Inequalities and Applications Volume 2009, Article ID 851360, 10 pages doi:10.1155/2009/851360 Research Article Some New Hilbert’s Type Inequalities Chang-Jian Zhao 1 and Wing-Sum Cheung 2 1 Department. Pang Some new inequalities similar to Hilbert’s type inequality involving series of nonnegative terms are established. Copyright q 2009 C J. Zhao and W S. Cheung. This is an open access article. given considerable attention to Hilbert’s type inequalities and their various generalizations. In particular, in 1, Pachpatte proved some new inequalities similar to Hilbert’s inequality 11, page
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