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 Aczél’s inequality and Popoviciu’s inequality by means of two classical inequalities, a unified improvement of Aczél’s inequality and Popoviciu’s inequality is given. As application, an integral inequality of Aczél-Popoviciu type is established. Copyright © 2007 Shanhe Wu. 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 1956, Aczél [1] proved the following result: a21 − n i=2 a2i b12 − n i=2 bi2 ≤ a1 b1 − n 2 ai bi , (1.1) i =2 where ai , bi (i = 1,2,...,n) are positive numbers such that a21 − ni=2 a2i >0 or b12 − ni=2 bi2 > 0. This inequality is called Aczél’s inequality. It is well known that Aczél’s inequality has important applications in the theory of functional equations in non-Euclidean geometry. In recent years, this inequality has attracted the interest of many mathematicians and has motivated a large number of research papers involving different proofs, various generalizations, improvements, and applications (see [2–11] and references therein). We state here a brief history on improvement of Aczél’s inequality. Popoviciu [12] first presented an exponential extension of Aczél’s inequality, as follows. 2 Journal of Inequalities and Applications Theorem 1.1. Let p > 0, q > 0, 1/ p + 1/q = 1, and let ai , bi (i = 1,2,...,n) be positive p p q q numbers such that a1 − ni=2 ai > 0 and b1 − ni=2 bi > 0. Then p a1 − n 1/ p p ai q b1 − i=2 n 1/q q bi ≤ a1 b1 − i=2 n ai bi . (1.2) i =2 Wu and Debnath [13] generalized inequality (1.2) in the following form. Theorem 1.2. Let p > 0, q > 0, and let ai , bi (i = 1,2,...,n) be positive numbers such that p p q q a1 − ni=2 ai > 0 and b1 − ni=2 bi > 0. Then p a1 − n 1/ p p q b1 − ai i=2 n 1/q q ≤ n1−min{ p bi −1 +q−1 ,1} i =2 a1 b1 − n ai bi . (1.3) i =2 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, and let ai , bi (i = 1,2,...,n) be positive p p q q numbers such that a1 − ni=2 ai > 0 and b1 − ni=2 bi > 0. Then p a1 − n 1/ p p ai i=2 q b1 − n 1/q q bi ≤ a1 b1 − i=2 n ai bi i =2 p q n ai bi a1 b1 − p − q max{ p, q,1} i=2 a1 b1 2 . (1.4) In this paper, we show a new sharp and generalized version of Popoviciu’s inequality, which is a unified improvement of Aczél’s inequality and Popoviciu’s inequality. In Section 4, the obtained result will be used to establish an integral inequality of AczélPopoviciu type. 2. Lemmas In order to prove the theorem in Section 3, we first introduce the following lemmas. Lemma 2.1 (generalized Hölder inequality [15, page 20]). Let ai j > 0, λ j ≥ 0 (i = 1,2,..., n, j = 1,2,...,m), and let λ1 + λ2 + · · · + λm = 1. Then m n j =1 λ j ai j i=1 ≥ n m λj ai j (2.1) i=1 j =1 with equality holding if and only if a11 /a1 j = a21 /a2 j = · · · = an1 /an j ( j = 2,3,...,m) for λ1 λ2 · · · λn = 0. Lemma 2.2 (mean value inequality [16, page 17]). Let xi > 0, λi > 0 (i = 1,2,...,n) and let λ1 + λ2 + · · · + λn = 1. Then n i=1 λ i xi ≥ n xiλi i =1 with equality holding if and only if x1 = x2 = · · · = xn . (2.2) Shanhe Wu 3 Lemma 2.3. Let p1 ≥ p2 ≥ · · · ≥ pm > 0, 1/ p1 + 1/ p2 + · · · + 1/ pm = 1, 0 < x j < 1 ( j = 1,2,...,m), and let xm+1 = x1 , pm+1 = p1 . Then m xj + j =1 m p j 1/ p j 1 − xj ≤ 1− j =1 p m 1 pj p j+1 2 x j − x j+1 2p1 j =1 p (2.3) p with equality holding if and only if x1 1 = x2 2 = · · · = xmm . Proof. From hypotheses in Lemma 2.3, it is easy to verify that 1 1 1 1 ≥ ≥ ··· ≥ ≥ > 0, pm pm−1 p2 p1 1 1 1 1 1 1 1 1 − ≥ 0, − ≥ 0,..., − ≥ 0, − ≥ 0, 2p2 2p1 2p3 2p2 2pm 2pm−1 2pm 2p1 1 1 1 1 1 1 1 1 1 1 + + − + + + − + ··· + + 2p1 2p1 2p2 2p1 2p2 2p2 2p3 2p2 2pm−2 2pm−2 + = 1 1 1 1 1 1 1 1 1 1 − + + + − + + + − 2pm−1 2pm−2 2pm−1 2pm−1 2pm 2pm−1 2p1 2p1 2pm 2p1 1 1 1 + + ··· + = 1. p1 p2 pm (2.4) Hence, by using Lemma 2.1 we obtain p p x1 1 + 1 − x2 2 1/2p1 p p x2 2 + 1 − x1 1 p2 p3 × x2 + 1 − x3 1/2p2 p 1/2p1 p p p 1/2p2 −1/2p1 p x2 2 + 1 − x2 2 x3 3 + 1 − x2 2 1/2p2 p x3 3 + 1 − x3 3 1/2p3 −1/2p2 .. . pm−2 pm−1 1/2pm−2 × xm−2 + 1 − xm−1 1/2pm−1 −1/2pm−2 pm−1 pm−2 1/2pm−2 p p × xm−1 + 1 − xm−2 xmm−−11 + 1 − xmm−−11 1/2pm−1 pm pm−1 pm 1/2pm−1 p p p 1/2pm −1/2pm−1 × xm−1 + 1 − xm xmm + 1 − xmm−−11 xm + 1 − xmm pm p1 1/2p1 p p 1/2p1 p p 1/2pm −1/2p1 × xm + 1 − x1 x1 1 + 1 − xmm xmm + 1 − xmm p1 /2p1 p2 /2p1 p2 /2p2 − p2 /2p1 p2 /2p2 pm−1 /2pm−2 pm−1 /2pm−1 − pm−1 /2pm−2 pm−1 /2pm−1 x2 x2 x2 · · · xm−1 xm−1 xm−1 ≥ x1 pm /2pm−1 pm /2pm − pm /2pm−1 pm /2p1 pm /2pm − pm /2p1 p1 /2p1 × xm xm xm xm x1 p1 1/2p1 p2 1/2p1 p2 1/2p2 −1/2p1 p 1/2p2 + 1 − x1 1 − x2 1 − x2 1 − x2 2 1/2pm−1 −1/2pm−2 1/2pm−1 pm−1 1/2pm−2 p p · · · 1 − xm−1 1 − xmm−−11 1 − xmm−−11 pm 1/2pm−1 p 1/2pm −1/2pm−1 p 1/2p1 p 1/2pm −1/2p1 × 1 − xm 1 − xmm 1 − xmm 1 − xmm p1 1/2p1 × 1 − x1 , (2.5) 4 Journal of Inequalities and Applications which is equivalent to p p 1 − x1 1 − x2 2 2 1/2p1 p p 1 − x2 2 − x3 3 2 1/2p2 p pm−1 pm 2 1/2pm−1 p 2 1/2p1 · · · 1 − (xm−1 − xm 1 − xmm − x1 1 p1 1/ p1 p 1/ p2 pm 1/ pm ≥ x1 x2 · · · xm + 1 − x1 1 − x2 2 · · · 1 − xm . (2.6) On the other hand, it follows from Lemma 2.2 that p p p 1 1 1 p 2 p 2 p 2 1 − x1 1 − x2 2 + 1 − x2 2 − x3 3 + · · · + 1 − xmm−−11 − xmm 2p1 2p2 2pm−1 p 1 1 1 1 1 p 2 ·1 + 1 − xmm − x1 1 + + + ··· + + 2p1 2p2 2p3 2pm−1 pm ≥ 1− p x1 1 ··· 1 − p2 2 1/2p1 − x2 p xmm−−11 1− p x2 2 pm 2 1/2pm−1 − xm p3 2 1/2p2 − x3 p p 1 − xmm − x1 1 2 1/2p1 (2.7) , this yields p p 1− x1 1 −x2 2 2 1/2p1 p p 1− x2 2 −x3 3 2 pm−1 p m 2 · · · 1− xm−1 − xm 1/2p2 1/2pm−1 p p 1 − xmm − x1 1 2 1/2p1 1 1 1 1 p1 1 p2 p 2 p 2 + + ··· + x − x2 2 − x − x3 3 − p1 p2 pm 2p1 1 2p2 2 1 pm−1 1 pm p 2 p 2 − ··· − xm−1 − xmm − xm − x1 1 2pm−1 2p1 p 1 p1 p2 2 p2 p3 2 p 2 p p 2 ≤ 1− x − x2 + x2 − x3 + · · · + xmm−−11 + xmm + xmm − x1 1 . 2p1 1 ≤ (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) p p p if and only if x1 1 = x2 2 = · · · = xmm , and thus we obtain the condition of equality in (2.3). The proof of Lemma 2.3 is complete. 3. Improvement of Aczél’s inequality and Popoviciu’s inequality pj Theorem 3.1. Let p1 ≥ p2 ≥ · · · ≥ pm > 0, 1/ p1 + 1/ p2 + · · · + 1/ pm = 1, ai j > 0, a1 j − n pj i=2 ai j > 0 (i = 1,2,...,n, j = 1,2,...,m), and let pm+1 = p1 , aim+1 = ai1 (i = 1,2,...,n). Then one has the following inequality: m j =1 pj a1 j − n i=2 1/ p j pj ai j m n m p p j j+1 m n ai j+1 a a · · · a1m ai j ≤ a1 j − ai j − 11 12 − pj p j+1 2p1 a1 j+1 j =1 i=2 j =1 j =1 i=2 a1 j 2 . (3.1) p pj p pj p pj Equality holds in (3.1) if and only if a111 /a1 j = a211 /a2 j = · · · = an11 /an j ( j = 2,3,...,m). Shanhe Wu 5 Proof. Since by hypotheses in Theorem 3.1 we have pj a1 j pj i =2 a i j − 0< 1/ p j n < 1 ( j = 1,2,...,m), p j 1/ p j a1 j pj it follows from Lemma 2.3, with a substitution x j = (a1 j − 1,2,...,m) in (2.3), that m a p j − n a p j 1/ p j 1j i=2 i j (3.2) n p j 1/ p pj j /(a )1/ p j 1j i =2 a i j ) n p j 1/ p j i=2 ai j pj pj a1 j a1 j j =1 n p j p j+1 p j+1 2 m pj a1 j+1 − ni=2 ai j+1 1 a 1 j − i =2 a i j ≤ 1− − , pj p j+1 2p1 j =1 a1 j a1 j+1 j =1 + m (j = (3.3) which is equivalent to m pj n a1 j − j =1 1/ p j pj ai j ≤ i=2 m m n a1 j − j =1 j =1 1/ p j pj − ai j i=2 p p j j+1 m n ai j+1 a11 a12 · · · a1m ai j − pj p j+1 2p1 a1 j+1 j =1 i=2 a1 j 2 , (3.4) pj pj p j+1 p j+1 where equality holds if and only if ( ni=2 ai j )/a1 j = ( ni=2 ai j+1 )/a1 j+1 ( j = 1,2,...,m), that pj pj p p is, if and only if a111 /a1 j = ( ni=2 ai11 )/( ni=2 ai j ) ( j = 2,3,...,m). On the other hand, using Lemma 2.1 gives m n j =1 1/ p j pj ai j ≥ i =2 n m ai j , (3.5) i=2 j =1 pj p pj p p pj where equality holds if and only if a211 /a2 j = a311 /a3 j = · · · = an11 /an j ( 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 pj pj pj p p p in (3.1) if and only if a111 /a1 j = a211 /a2 j = · · · = an11 /an j ( j = 2,3,...,m). This completes the proof of Theorem 3.1. As a consequence of Theorem 3.1, putting m = 2, p1 = p, p2 = q, ai1 = ai , ai2 = bi (i = 1,2,...,n) in (3.1), we get the following. Corollary 3.2. Let p ≥ q > 0, 1/ p + 1/q = 1, and let ai , bi (i = 1,2,...,n) be positive num p p q q bers such that a1 − ni=2 ai > 0 and b1 − ni=2 bi > 0. Then p a1 − n i=2 1/ p p ai q b1 − n 1/q q bi ≤ a1 b1 − i=2 n i =2 p q p q ai bi p q n bi a1 b1 ai − − p i=2 a1p b1q p q with equality holding if and only if a1 /b1 = a2 /b2 = · · · = an /bn . 2 (3.6) 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, and let ai , bi (i = 1,2,...,n) be positive p p q q numbers such that a1 − ni=2 ai > 0 and b1 − ni=2 bi > 0. Then p a1 − n 1/ p p ai q b1 − i=2 n 1/q q bi ≤ a1 b1 − i =2 n i=2 p q p ai bi p q n ai bi a1 b1 − p − q max{ p, q} i=2 a1 b1 q p 2 , (3.7) q with equality holding if and only if a1 /b1 = a2 /b2 = · · · = an /bn . Obviously, inequalities (3.1), (3.6), and (3.7) are the improvement of Aczél’s inequality and Popoviciu’s inequality. 4. Integral version of Aczél-Popoviciu-type inequality As application of Theorem 3.1, we establish here an interesting integral inequality of Aczél-Popoviciu type. Theorem 4.1. Let p1 ≥ p2 ≥ · · · ≥ pm > 0, 1/ p1 + 1/ p2 + · · · + 1/ pm = 1, B j > 0 ( j = pj 1,2,...,m), let f j be positive Riemann integrable functions on [a,b] such that B j − b pj a f j (x)dx > 0 for all j = 1,2,...,m, and let Bm+1 = B1 , pm+1 = p1 , fm+1 = f1 . Then one has the following inequality: m pj Bj − j =1 ≤ m b a 1/ p j pj f j (x)dx Bj − b m j =1 2 p j+1 m b f p j (x) f j+1 (x) B1 B2 · · · Bm j f j (x) dx − dx . pj − p j+1 2p1 a Bj B j+1 j =1 j =1 (4.1) a Proof. For any positive integer n, we choose an equidistant partition of [a,b] as a < a+ b−a b−a b−a < ··· < a + i < ··· < a + (n − 1) < b, n n n b−a Δxi = , i = 1,2,...,n. n pj b pj a f j (x)dx Since the hypothesis B j − pj B j − lim n→∞ n pj fj i =1 a+ (4.2) > 0 ( j = 1,2,...,m) implies that i(b − a) b − a > 0 ( j = 1,2,...,m), n n (4.3) there exists a positive integer N such that pj Bj − n i=1 pj fj a+ i(b − a) b − a > 0 ∀n > N, j = 1,2,...,m. n n (4.4) Shanhe Wu 7 Applying Theorem 3.1, one obtains for any n > N the following inequality: m pj Bj − j =1 n pj fj i=1 ≤ m Bj − j =1 i(b − a) b − a a+ n n 1/ p j n m i(b − a) fj a + n j =1 i =1 1/ p1 +1/ p2 +···+1/ pm b−a n m n i(b − a) b − a B1 B2 · · · Bm 1 p j − a+ pj f j 2p1 n n j =1 i=1 B j − 1 p j+1 f j+1 a + p j+1 B j+1 i(b − a) b − a n n (4.5) 2 . Note that 1/ p1 + 1/ p2 + · · · + 1/ pm = 1, the above inequality can be transformed to m pj Bj − j =1 n pj fj i=1 ≤ m Bj − j =1 i(b − a) b − a a+ n n n m i =1 fj a + j =1 1/ p j i(b − a) n b−a n m n i(b − a) B1 B2 · · · Bm 1 p j a+ − pj f j 2p1 n j =1 i=1 B j − 1 p j+1 B j+1 p j+1 f j+1 i(b − a) a+ n pj pj (4.6) b−a n 2 , p j+1 p j+1 where equality holds if and only if f j (a + i(b − a)/n)/B j = f j+1 (a + i(b − a)/n)/B j+1 for all i = 1,2,...,n ( j = 1,2,...,m). In view of the hypotheses that f j are positive Riemann integrable functions on [a,b] pj and p j > 0 ( j = 1,2,...,m), we conclude that mj=1 f j and f j ( j = 1,2,...,m) are also 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 p1 ≥ p2 ≥ · · · ≥ pm > 0, 1/ p1 + 1/ p2 + · · · + 1/ pm = 1, B j > pj b 0 ( j = 1,2,...,m), suppose also that f j ∈ L p j [a,b], B j − a | f j (x)| p j dx > 0 for all j = 1,2,...,m, let Bm+1 = B1 , pm+1 = p1 , fm+1 = f1 . Then the following inequality holds true: m pj Bj − j =1 ≤ m j =1 b a f j (x) p j dx 1/ p j 2 b m b m f j (x) p j f j+1 (x) p j+1 B1 B2 · · · Bm B j− f j (x) dx − − dx pj p j+1 a j =1 2p1 j =1 a Bj B j+1 (4.7) 8 Journal of Inequalities and Applications pj p j+1 with equality holding if and only if | f j (x)| p j /B j = | f j+1 (x)| p j+1 /B j+1 ( j = 1,2,...,m) almost everywhere on [a,b]. As a consequence of Theorem 4.1, putting m = 2, p1 = p, p2 = q, B1 = A, B2 = B, f1 = f , f2 = g in (4.1), we obtain the following. Corollary 4.4. Let p ≥ q > 0, 1/ p + 1/q = 1, A > 0, B > 0, and let f , g be positive Riemann b b integrable functions on [a,b] such that A p − a f p (x)dx > 0 and B q − a g q (x)dx > 0. Then Ap − b a 1/ p f p (x)dx ≤ AB − b a Bq − b a 1/q g q (x)dx AB f (x)g(x)dx − p b f p (x) g q (x) − dx Ap Bq a (4.8) 2 . 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, and let f , g be positive b b Riemann integrable functions on [a,b] such that A p − a f p (x)dx > 0 and B q − a g q (x)dx > 0. Then p A − 1/ p b a p f (x)dx ≤ AB − b a q B − b a 1/q q g (x)dx AB f (x)g(x)dx − max{ p, q} b a f p (x) g q (x) − dx Ap Bq (4.9) 2 . Acknowledgment The author would like to express hearty thanks to the anonymous referees for valuable comments on this paper. References [1] J. Aczél, “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. Matić, and J. 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[8] R. Bellman, “On an inequality concerning an indefinite form,” The American Mathematical Monthly, vol. 63, no. 2, pp. 108–109, 1956. Shanhe Wu 9 [9] P. M. Vasić and J. E. Pečarić, “On the Jensen inequality for monotone functions,” Analele Universităţii din Timişoara. Seria Matematică-Informatică, vol. 17, no. 1, pp. 95–104, 1979. [10] J. C. Kuang, Applied Inequalities, Hunan Education Press, Changsha, China, 2nd edition, 1993. [11] D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, vol. 61, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. [12] T. Popoviciu, “On an inequality,” Gazeta Matematica si Fizica. Seria A, vol. 11 (64), pp. 451–461, 1959 (Romanian). [13] S. Wu and L. Debnath, “Generalizations of Aczél’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 Aczél’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. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1952. Shanhe Wu: Department of Mathematics, Longyan College, Longyan, Fujian 364012, China Email address: wushanhe@yahoo.com.cn