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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 72173, 9 pages doi:10.1155/2007/72173 Research Article Improvement of Aczél’s Inequality and Popoviciu’s Inequality Shanhe Wu Received 30 December 2006; Accepted 24 April 2007 Recommended by Laszlo I. Losonczi We generalize and sharpen Acz ´ el’s inequality and Popoviciu’s inequality by means of two classical inequalities, a unified improvement of Acz ´ el’s inequality and Popoviciu’s inequality is given. As application, an integral inequality of Acz ´ el-Popoviciu type is es- tablished. Copyright © 2007 Shanhe Wu. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In 1956, Acz ´ el [1] proved the following result:  a 2 1 − n  i=2 a 2 i  b 2 1 − n  i=2 b 2 i  ≤  a 1 b 1 − n  i=2 a i b i  2 , (1.1) where a i , b i (i=1,2, ,n) are positive numbers such that a 2 1 −  n i =2 a 2 i >0orb 2 1 −  n i =2 b 2 i > 0. This inequality is called Acz ´ el’s inequality. It is well known that Acz ´ el’s inequality has important applications in the theory of functional equations in non-Euclidean geometry. In recent years, this inequality has at- tracted the interest of many mathematicians and has motivated a large number of re- search papers involving different proofs, various generalizations, improvements, and ap- plications (see [2–11] and references therein). We state here a brief history on improve- ment of Acz ´ el’s inequality. Popoviciu [12] first presented an exponential extension of Acz ´ el’s inequality, as fol- lows. 2 Journal of Inequalities and Applications Theorem 1.1. Let p>0, q>0, 1/p+1/q = 1,andleta i , b i (i = 1,2, ,n) be positive numbers such that a p 1 −  n i =2 a p i > 0 and b q 1 −  n i =2 b q i > 0. Then  a p 1 − n  i=2 a p i  1/p  b q 1 − n  i=2 b q i  1/q ≤ a 1 b 1 − n  i=2 a i b i . (1.2) Wu and Debnath [13] generalized inequality (1.2) in the following form. Theorem 1.2. Let p>0, q>0,andleta i , b i (i = 1,2, ,n) be positive numbers such that a p 1 −  n i =2 a p i > 0 and b q 1 −  n i =2 b q i > 0. Then  a p 1 − n  i=2 a p i  1/p  b q 1 − n  i=2 b q i  1/q ≤ n 1−min{p −1 +q −1 ,1} a 1 b 1 − n  i=2 a i b i . (1.3) In a recent paper [14], Wu established a sharp and generalized version of Popoviciu’s inequality as follows. Theorem 1.3. Let p>0, q>0, 1/p+1/q ≥ 1,andleta i , b i (i = 1,2, ,n) be positive numbers such that a p 1 −  n i =2 a p i > 0 and b q 1 −  n i =2 b q i > 0. Then  a p 1 − n  i=2 a p i  1/p  b q 1 − n  i=2 b q i  1/q ≤ a 1 b 1 −  n  i=2 a i b i  − a 1 b 1 max{p,q,1}  n  i=2  a p i a p 1 − b q i b q 1  2 . (1.4) In this paper, we show a new sharp and generalized version of Popoviciu’s inequal- ity, which is a unified improvement of Acz ´ el’s inequality and Popoviciu’s inequality. In Section 4, the obtained result will be used to establish an integr al inequality of Acz ´ el- Popoviciu type. 2. Lemmas In order to prove the theorem in Section 3, we first introduce the following lemmas. Lemma 2.1 (generalized H ¨ older inequality [15, page 20]). Let a ij > 0, λ j ≥ 0(i = 1,2, , n, j = 1,2, ,m),andletλ 1 + λ 2 + ···+ λ m = 1. Then m  j=1  n  i=1 a ij  λ j ≥ n  i=1 m  j=1 a λ j ij (2.1) with equality holding if and only if a 11 /a 1j = a 21 /a 2j = ··· = a n1 /a nj ( j = 2, 3, , m) for λ 1 λ 2 ···λ n = 0. Lemma 2.2 (mean value inequality [16, page 17]). Let x i > 0, λ i > 0(i = 1,2, ,n) and let λ 1 + λ 2 + ···+ λ n = 1. Then n  i=1 λ i x i ≥ n  i=1 x λ i i (2.2) with equality holding if and only if x 1 = x 2 =···=x n . Shanhe Wu 3 Lemma 2.3. Le t p 1 ≥ p 2 ≥ ··· ≥ p m > 0, 1/p 1 +1/p 2 + ···+1/p m = 1, 0 <x j < 1(j = 1,2, ,m),andletx m+1 = x 1 , p m+1 = p 1 . Then m  j=1 x j + m  j=1  1 − x p j j  1/p j ≤ 1 − 1 2p 1 m  j=1  x p j j − x p j+1 j+1  2 (2.3) with equality holding if and only if x p 1 1 = x p 2 2 =···=x p m m . Proof. From hypotheses in Lemma 2.3,itiseasytoverifythat 1 p m ≥ 1 p m−1 ≥···≥ 1 p 2 ≥ 1 p 1 > 0, 1 2p 2 − 1 2p 1 ≥ 0, 1 2p 3 − 1 2p 2 ≥ 0, , 1 2p m − 1 2p m−1 ≥ 0, 1 2p m − 1 2p 1 ≥ 0, 1 2p 1 + 1 2p 1 +  1 2p 2 − 1 2p 1  + 1 2p 2 + 1 2p 2 +  1 2p 3 − 1 2p 2  + ···+ 1 2p m−2 + 1 2p m−2 +  1 2p m−1 − 1 2p m−2  + 1 2p m−1 + 1 2p m−1 +  1 2p m − 1 2p m−1  + 1 2p 1 + 1 2p 1 +  1 2p m − 1 2p 1  = 1 p 1 + 1 p 2 + ···+ 1 p m = 1. (2.4) Hence, by using Lemma 2.1 we obtain  x p 1 1 +  1 − x p 2 2  1/2p 1  x p 2 2 +  1 − x p 1 1  1/2p 1  x p 2 2 +  1 − x p 2 2  1/2p 2 −1/2p 1 ×  x p 2 2 +  1 − x p 3 3  1/2p 2  x p 3 3 +  1 − x p 2 2  1/2p 2  x p 3 3 +  1 − x p 3 3  1/2p 3 −1/2p 2 . . . ×  x p m−2 m−2 +  1 − x p m−1 m−1  1/2p m−2 ×  x p m−1 m−1 +  1 − x p m−2 m−2  1/2p m−2  x p m−1 m−1 +  1 − x p m−1 m−1  1/2p m−1 −1/2p m−2 ×  x p m−1 m−1 +  1 − x p m m  1/2p m−1  x p m m +  1 − x p m−1 m−1  1/2p m−1  x p m m +  1 − x p m m  1/2p m −1/2p m−1 ×  x p m m +  1 − x p 1 1  1/2p 1  x p 1 1 +  1 − x p m m  1/2p 1  x p m m +  1 − x p m m  1/2p m −1/2p 1 ≥ x p 1 /2p 1 1 x p 2 /2p 1 2 x p 2 /2p 2 −p 2 /2p 1 2 x p 2 /2p 2 2 ···x p m−1 /2p m−2 m−1 x p m−1 /2p m−1 −p m−1 /2p m−2 m−1 x p m−1 /2p m−1 m−1 × x p m /2p m−1 m x p m /2p m −p m /2p m−1 m x p m /2p 1 m x p m /2p m −p m /2p 1 m x p 1 /2p 1 1 +  1 − x p 1 1  1/2p 1  1 − x p 2 2  1/2p 1  1 − x p 2 2  1/2p 2 −1/2p 1  1 − x p 2 2  1/2p 2 ···  1 − x p m−1 m−1  1/2p m−2  1 − x p m−1 m−1  1/2p m−1 −1/2p m−2  1 − x p m−1 m−1  1/2p m−1 ×  1 − x p m m  1/2p m−1  1 − x p m m  1/2p m −1/2p m−1  1 − x p m m  1/2p 1  1 − x p m m  1/2p m −1/2p 1 ×  1 − x p 1 1  1/2p 1 , (2.5) 4 Journal of Inequalities and Applications which is equivalent to  1 −  x p 1 1 − x p 2 2  2  1/2p 1  1 −  x p 2 2 − x p 3 3  2  1/2p 2 ···  1 − (x p m−1 m−1 − x p m m  2  1/2p m−1  1 −  x p m m − x p 1 1  2  1/2p 1 ≥ x 1 x 2 ···x m +  1 − x p 1 1  1/p 1  1 − x p 2 2  1/p 2 ···  1 − x p m m  1/p m . (2.6) On the other hand, it follows from Lemma 2.2 that 1 2p 1  1 −  x p 1 1 − x p 2 2  2  + 1 2p 2  1 −  x p 2 2 − x p 3 3  2  + ···+ 1 2p m−1  1 −  x p m−1 m−1 − x p m m  2  + 1 2p 1  1 −  x p m m − x p 1 1  2  +  1 2p 2 + 1 2p 3 + ···+ 1 2p m−1 + 1 p m  · 1 ≥  1 −  x p 1 1 − x p 2 2  2  1/2p 1  1 −  x p 2 2 − x p 3 3  2  1/2p 2 ···  1 −  x p m−1 m−1 − x p m m  2  1/2p m−1  1 −  x p m m − x p 1 1  2  1/2p 1 , (2.7) this y ields  1−  x p 1 1 −x p 2 2  2  1/2p 1  1−  x p 2 2 −x p 3 3  2  1/2p 2 ···  1−  x p m−1 m−1 −x p m m  2  1/2p m−1  1−  x p m m −x p 1 1  2  1/2p 1 ≤  1 p 1 + 1 p 2 + ···+ 1 p m  − 1 2p 1  x p 1 1 − x p 2 2  2 − 1 2p 2  x p 2 2 − x p 3 3  2 −···− 1 2p m−1  x p m−1 m−1 − x p m m  2 − 1 2p 1  x p m m − x p 1 1  2 ≤ 1 − 1 2p 1  x p 1 1 − x p 2 2  2 +  x p 2 2 − x p 3 3  2 + ···+  x p m−1 m−1 + x p m m  2 +  x p m m − x p 1 1  2  . (2.8) Combining inequalities (2.6)and(2.8) leads to inequality (2.3). In addition, from Lemmas 2.1 and 2.2, we can easily deduce that the equality holds in both (2.6)and(2.8) if and only if x p 1 1 = x p 2 2 =···=x p m m , and thus we obtain the condition of equality in (2.3). The proof of Lemma 2.3 is complete.  3. Improvement of Acz ´ el’s inequality and Popov i ciu’s inequality Theorem 3.1. Let p 1 ≥ p 2 ≥ ··· ≥ p m > 0, 1/p 1 +1/p 2 + ···+1/p m = 1, a ij > 0, a p j 1j −  n i =2 a p j ij > 0(i = 1, 2, , n, j = 1, 2, , m),andletp m+1 = p 1 , a im+1 = a i1 (i = 1,2, ,n). Then one has the following inequality: m  j=1  a p j 1j − n  i=2 a p j ij  1/p j ≤ m  j=1 a 1j − n  i=2 m  j=1 a ij − a 11 a 12 ···a 1m 2p 1 m  j=1  n  i=2  a p j ij a p j 1j − a p j+1 ij+1 a p j+1 1j+1  2 . (3.1) Equality holds in (3.1)ifandonlyifa p 1 11 /a p j 1j = a p 1 21 /a p j 2j =···=a p 1 n1 /a p j nj ( j = 2,3, ,m). Shanhe Wu 5 Proof. Since by hypotheses in Theorem 3.1 we have 0 <  a p j 1j −  n i =2 a p j ij  1/p j  a p j 1j  1/p j < 1(j = 1,2, ,m), (3.2) it follows from Lemma 2.3, with a substitution x j = (a p j 1j −  n i =2 a p j ij ) 1/p j /(a p j 1j ) 1/p j ( j = 1,2, ,m)in(2.3), that m  j=1  a p j 1j −  n i =2 a p j ij a p j 1j  1/p j + m  j=1   n i =2 a p j ij a p j 1j  1/p j ≤ 1 − 1 2p 1 m  j=1  a p j 1j −  n i =2 a p j ij a p j 1j − a p j+1 1j+1 −  n i =2 a p j+1 ij+1 a p j+1 1j+1  2 , (3.3) which is equivalent to m  j=1  a p j 1j − n  i=2 a p j ij  1/p j ≤ m  j=1 a 1j − m  j=1  n  i=2 a p j ij  1/p j − a 11 a 12 ···a 1m 2p 1 m  j=1  n  i=2  a p j ij a p j 1j − a p j+1 ij+1 a p j+1 1j+1  2 , (3.4) where equality holds if and only if (  n i =2 a p j ij )/a p j 1j = (  n i =2 a p j+1 ij+1 )/a p j+1 1j+1 ( j = 1,2, ,m), that is, if and only if a p 1 11 /a p j 1j = (  n i =2 a p 1 i1 )/(  n i =2 a p j ij )(j = 2, 3, ,m). On the other hand, using Lemma 2.1 gives m  j=1  n  i=2 a p j ij  1/p j ≥ n  i=2 m  j=1 a ij , (3.5) where equality holds if and only if a p 1 21 /a p j 2j = a p 1 31 /a p j 3j =···=a p 1 n1 /a p j nj ( j = 2,3, ,m). Combining inequalities (3.4)and(3.5) leads to the desired inequality (3.1). By means of the conditions of equality in (3.4)and(3.5), it is easy to conclude that there is equality in (3.1)ifandonlyifa p 1 11 /a p j 1j = a p 1 21 /a p j 2j = ··· = a p 1 n1 /a p j nj ( j = 2,3, ,m). This completes the proof of Theorem 3.1.  As a consequence of Theorem 3.1,puttingm = 2, p 1 = p, p 2 = q, a i1 = a i , a i2 = b i (i = 1,2, ,n)in(3.1), we get the fol low ing. Corollary 3.2. Let p ≥ q>0, 1/p+1/q = 1,andleta i , b i (i = 1,2, ,n) be positive num- bers such that a p 1 −  n i =2 a p i > 0 and b q 1 −  n i =2 b q i > 0. Then  a p 1 − n  i=2 a p i  1/p  b q 1 − n  i=2 b q i  1/q ≤ a 1 b 1 −  n  i=2 a i b i  − a 1 b 1 p  n  i=2  a p i a p 1 − b q i b q 1  2 (3.6) with equality holding if and only if a p 1 /b q 1 = a p 2 /b q 2 =···=a p n /b q n . 6 Journal of Inequalities and Applications A simple application of Corollary 3.2 yields the following sharp version of Popoviciu’s inequality. Corollary 3.3. Let p>0, q>0, 1/p+1/q = 1,andleta i , b i (i = 1, 2, , n) be positive numbers such that a p 1 −  n i =2 a p i > 0 and b q 1 −  n i =2 b q i > 0. Then  a p 1 − n  i=2 a p i  1/p  b q 1 − n  i=2 b q i  1/q ≤ a 1 b 1 −  n  i=2 a i b i  − a 1 b 1 max{p,q}  n  i=2  a p i a p 1 − b q i b q 1  2 , (3.7) with equality holding if and only if a p 1 /b q 1 = a p 2 /b q 2 =···=a p n /b q n . Obviously, inequalities (3.1), (3.6), and (3.7) are the improvement of Acz ´ el’s inequality and Popoviciu’s inequality. 4. Integral version of Acz ´ el-Popoviciu-type inequality As application of Theorem 3.1, we establish here an interesting integral inequality of Acz ´ el-Popoviciu type. Theorem 4.1. Let p 1 ≥ p 2 ≥ ··· ≥ p m > 0, 1/p 1 +1/p 2 + ··· +1/p m = 1, B j > 0(j = 1,2, ,m),let f j be positive Riemann integrable functions on [a,b] such that B p j j −  b a f p j j (x) dx > 0 for all j = 1,2, ,m,andletB m+1 = B 1 , p m+1 = p 1 , f m+1 = f 1 .Thenone has the following inequality: m  j=1  B p j j −  b a f p j j (x)dx  1/p j ≤ m  j=1 B j −  b a  m  j=1 f j (x)  dx − B 1 B 2 ···B m 2p 1 m  j=1   b a  f p j j (x) B p j j − f p j+1 j+1 (x) B p j+1 j+1  dx  2 . (4.1) Proof. For any positive integer n, we choose an equidistant partition of [a,b]as a<a+ b − a n < ···<a+ b − a n i< ···<a+ b − a n (n − 1) <b, Δx i = b − a n , i = 1,2, ,n. (4.2) Since the hypothesis B p j j −  b a f p j j (x) dx > 0(j = 1,2, ,m) implies that B p j j − lim n→∞ n  i=1 f p j j  a + i(b − a) n  b − a n > 0(j = 1,2, ,m), (4.3) there exists a positive integer N such that B p j j − n  i=1 f p j j  a + i(b − a) n  b − a n > 0 ∀n>N, j = 1,2, ,m. (4.4) Shanhe Wu 7 Applying Theorem 3.1, one obtains for any n>N the following inequalit y: m  j=1  B p j j − n  i=1 f p j j  a + i(b − a) n  b − a n  1/p j ≤ m  j=1 B j − n  i=1  m  j=1 f j  a + i(b − a) n    b − a n  1/p 1 +1/p 2 +···+1/p m − B 1 B 2 ···B m 2p 1 m  j=1  n  i=1  1 B p j j f p j j  a + i(b − a) n  b − a n − 1 B p j+1 j+1 f p j+1 j+1  a + i(b − a) n  b − a n  2 . (4.5) Note that 1/p 1 +1/p 2 + ···+1/p m = 1, the above inequality can be transformed to m  j=1  B p j j − n  i=1 f p j j  a + i(b − a) n  b − a n  1/p j ≤ m  j=1 B j − n  i=1  m  j=1 f j  a + i(b − a) n    b − a n  − B 1 B 2 ···B m 2p 1 m  j=1  n  i=1  1 B p j j f p j j  a + i(b − a) n  − 1 B p j+1 j+1 f p j+1 j+1  a + i(b − a) n   b − a n  2 , (4.6) where equality holds if and only if f p j j (a + i(b − a)/n)/B p j j = f p j+1 j+1 (a + i(b − a)/n)/B p j+1 j+1 for all i = 1,2, ,n ( j = 1,2, ,m). In view of the hypotheses that f j are positive Riemann integ rable functions on [a,b] and p j > 0(j = 1,2, ,m), we conclude that  m j =1 f j and f p j j ( j = 1,2, ,m)arealso integrable on [a,b]. Passing the limit as n →∞in both sides of inequality (4.6), we obtain the inequality (4.1). The proof of Theorem 4.1 is complete.  Remark 4.2. Motivated by the proof of Theorem 4.1, we propose here a conjecture. Conjecture 4.3. Suppose that p 1 ≥ p 2 ≥··· ≥ p m > 0, 1/p 1 +1/p 2 + ···+1/p m = 1, B j > 0(j = 1,2, ,m), suppose also that f j ∈ L p j [a,b], B p j j −  b a | f j (x)| p j dx > 0forallj = 1,2, ,m,letB m+1 = B 1 , p m+1 = p 1 , f m+1 = f 1 . Then the following inequality holds true: m  j=1  B p j j −  b a   f j (x)   p j dx  1/p j ≤ m  j=1 B j −  b a  m  j=1   f j (x)    dx− B 1 B 2 ···B m 2p 1 m  j=1   b a    f j (x)   p j B p j j −   f j+1 (x)   p j+1 B p j+1 j+1  dx  2 (4.7) 8 Journal of Inequalities and Applications with equality holding if and only if | f j (x)| p j /B p j j =|f j+1 (x)| p j+1 /B p j+1 j+1 ( j = 1,2, ,m)al- most everywhere on [a,b]. As a consequence of Theorem 4.1,puttingm = 2, p 1 = p, p 2 = q, B 1 = A, B 2 = B, f 1 = f , f 2 = g in (4.1), we obtain the following. Corollary 4.4. Let p ≥ q>0, 1/p+1/q = 1, A>0, B>0,andlet f , g be positive Riemann integrable functions on [a,b] such that A p −  b a f p (x)dx > 0 and B q −  b a g q (x)dx > 0. Then  A p −  b a f p (x) dx  1/p  B q −  b a g q (x) dx  1/q ≤ AB −  b a f (x)g(x)dx − AB p   b a  f p (x) A p − g q (x) B q  dx  2 . (4.8) Further , from Corollary 4.4 we have the following. Corollary 4.5. Let p>0, q>0, 1/p +1/q = 1, A>0, B>0,andlet f , g be positive Riemann integrable functions on [a, b] such that A p −  b a f p (x)dx > 0 and B q −  b a g q (x) dx > 0. Then  A p −  b a f p (x) dx  1/p  B q −  b a g q (x) dx  1/q ≤ AB −  b a f (x)g(x)dx − AB max{p,q}   b a  f p (x) A p − g q (x) B q  dx  2 . (4.9) Acknowledgment The author would like to express hearty thanks to the anonymous referees for valuable comments on this paper. References [1] J. Acz ´ el, “Some general methods in the theory of functional equations in one variable. New applications of functional equations,” Uspekhi Matematicheskikh Nauk (N.S.), vol. 11, no. 3(69), pp. 3–68, 1956 (Russian). [2] Y.J.Cho,M.Mati ´ c, and J. Pe ˇ cari ´ c, “Improvements of some inequalities of Acz ´ el’s type,” Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 226–240, 2001. [3] X H. Sun, “Acz ´ el-Chebyshev type inequality for positive linear functions,” Journal of Mathe- matical Analysis and Applications, vol. 245, no. 2, pp. 393–403, 2000. [4] L.LosoncziandZ.P ´ ales, “Inequalities for indefinite forms,” Journal of Mathematical Analysis and Applications, vol. 205, no. 1, pp. 148–156, 1997. [5] A. M. Mercer, “Extensions of Popoviciu’s inequality using a general method,” Journal of Inequal- ities in Pure and Applied Mathematic s , vol. 4, no. 1, Article 11, pp. 4 pages, 2003. [6] V. Mascioni, “A note on Acz ´ el type inequalities,” Journal of Inequalities in Pure and Applied Math- ematic s, vol. 3, no. 5, Article 69, pp. 5 pages, 2002. [7] S. S. Dragomir and B. 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Debnath, “Generalizations of Acz ´ el’s inequality and Popoviciu’s inequality,” Indian Journal of Pure and Applied Mathematics, vol. 36, no. 2, pp. 49–62, 2005. [14] S. Wu, “A further generalization of Acz ´ el’s inequality and Popoviciu’s inequality,” Mathematical Inequalities and Application, vol. 10, no. 3, 2007. [15] E. F. Beckenbach and R. Bellman, Inequalities, Springer, New York, NY, USA, 1983. [16] G. H. Hardy, J. E. Littlewood, and G. P ´ olya, Inequalities, Cambridge University Press, Cam- bridge, UK, 2nd edition, 1952. Shanhe Wu: Department of Mathematics, Longyan College, Longyan, Fujian 364012, China Email address: wushanhe@yahoo.com.cn . Corporation Journal of Inequalities and Applications Volume 2007, Article ID 72173, 9 pages doi:10.1155/2007/72173 Research Article Improvement of Aczél’s Inequality and Popoviciu’s Inequality Shanhe. Losonczi We generalize and sharpen Acz ´ el’s inequality and Popoviciu’s inequality by means of two classical inequalities, a unified improvement of Acz ´ el’s inequality and Popoviciu’s inequality is. a 1 b 1 −  n  i=2 a i b i  − a 1 b 1 max{p,q,1}  n  i=2  a p i a p 1 − b q i b q 1  2 . (1.4) In this paper, we show a new sharp and generalized version of Popoviciu’s inequal- ity, which is a unified improvement of Acz ´ el’s inequality and Popoviciu’s inequality. In Section 4, the obtained