Research Article A Converse of Minkowski s Type Inequalities

Hindawi Publishing Corporation Journal of Inequalities and Applications Volue 2010, Article ID 461215, 9 pages doi:10.1155/2010/461215 Research Article A Converse of Minkowski s Type Inequalities Roeo Meštrović 1 and David Kalaj 2 1 Maritie Faculty, University of Montenegro, Dobrota 36, 85330 Kotor, Montenegro 2 Faculty of Natural Sciences and Matheatics, University of Montenegro, Džordža Vašingtona BB, 81000 Podgorica, Montenegro Correspondence should be addressed to Roeo Meštrović, roeo@ac.e Received 6 August 2010; Accepted 20 October 2010 Acadeic Editor: Jong Ki Copyright q 2010 R. Meštrović and D. Kalaj. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. We forulate and prove a converse for a generalization of the classical Minkowski s inequality. The case when 0 <p<1 is also considered. Applying the sae technique, we obtain an analog converse theore for integral Minkowski s type inequality. 1. Introduction If p>1, a i 0, and b i 0 i 1,...,n are real nubers, then by the classical Minkowski s inequality n n n a i b i p a p i b p i. 1.1 This inequality was published by Minkowski 1, pages 115 117 hundred years ago in his faous book Geoetrie der Zahlen. It is also known see 2 that for 0 <p<1 the above inequality is satisfied with instead of. Many extensions and generalizations of Minkowski s inequality can be found in 2, 3. We want to point out the following inequality: p a ij, 1.2

2 Journal of Inequalities and Applications where p>1anda ij 0 i 1,...,; j 1,...,n are real nubers. Furtherore, if 0 < p<1, then the inequality 1.2 is satisfied with instead of 2, Theore 24, page 30. In both cases, equality holds if and only if all coluns a 1j,a 2j,...,a j, j 1, 2,...,n,are proportional. An extension of inequality 1.2 was forulated by Ingha and Jessen see 2, pages 31-32. In 1948, Tôyaa 4 published a converse of the inequality of Ingha and Jessen see also a recent paper 5 for a weighted version of Tôyaa s inequality.naely,tôyaa showed that if 0 <q<pand a ij 0 i 1,...,; j 1,...,n are real nubers, then q/p 1/q in, n 1/q p/q. 1.3 The ain result of this paper gives a converse of inequality 1.2. On the other hand, our result ay be regarded as a nonsyetric analogue of the above inequality, and it is given as follows. Theore 1.1. Let p>0, q>0, and a ij 0 i 1,...,; j 1,...,n be real nubers. Then for p 1 we have C p/q, 1.4 where C is a positive constant given by 1 1/q if 1 p q, C in, n 1/q 1 1/q if 1 q<p, 1 if 0 <q 1 p. 1.5 If 0 <p<1, then K p/q, 1.6 where K is a positive constant given by 1 1/q if 0 <q p<1, K in, n 1/q 1 1/q if 0 <p<q<1, 1 if 0 <p<1 q. 1.7

Journal of Inequalities and Applications 3 Inequality 1.4 with 1 p q and inequality 1.6 with 0 <q p<1 are sharp for all and n, and they are attained for a ij a, i 1,...,, j 1,...,n.If n, then inequality 1.4 is sharp in the cases when 1 q<pand 0 <q 1 p. In both cases the equalities are attained for a, if i j, a ij 0, if i / j. 1.8 When n, the equalities in 1.6 concerned with 0 <p<q<1 and 0 <p<1 q are also attained for previously defined values a ij. Reark 1.2. Note that, proceeding as in the proof of Theore 1.1, we can prove siilar inequalities to 1.4 and 1.6 with n instead of n on the left-hand side of these inequalities. For exaple, such an inequality concerning the case when 1 q<p i.e., 1.4 is n n 1 p/q. 1.9 The above inequality is sharp if n, but it is not in spirit of a converse of Minkowski s type inequality. The following consequence of Theore 1.1 for 2andq 2 can be viewed as a converse of Minkowski s inequality 1.1. Corollary 1.3. Let n 1, p>0, and let a j 0, b j 0 j 1,...,n be real nubers. Then for p 1 a p j b p j 2 1 in{1/2,} p/2 a 2 j b2 j. 1.10 If 0 <p<1, then a p j v p j 2 1 p/2 a 2 j b2 j. 1.11 Reark 1.4. It is well known that Minkowski s inequality is also true for coplex sequences as well. More precisely, if p 1andu i, v i i 1,...,n are arbitrary coplex nubers, then uj v j p uj p vj p. 1.12

4 Journal of Inequalities and Applications Note that the above inequality with u j a j R and v j ib j, b j R, for each j 1, 2,...,n, becoes p/2 a 2 j b2 j a p j v p j. 1.13 We see that the first inequality of Corollary 1.3 ay be actually regarded as a converse of the previous inequality. 2. Proof of Theore 1.1 Lea 2.1 see 2, page 26. If u 1,u 2,...u k,s,r are nonnegative real nubers and 0 <s<r, then u s 1 us 2 1/s us k u r 1 ur 2 1/r. ur k 2.1 Proof of Theore 1.1. In our proof we often use the well-known fact that the scale of power eans is nondecreasing see 2. More precisely, if a 1,a 2,...,a k are nonnegative integers and 0 <α β<, then k aα i k 1/α k aβ i k 1/β. 2.2 In all the cases, for each i 1, 2,...,, we denote that a i :. 2.3 We will consider all the six cases related to the inequalities 1.4 and 1.6. Case 1 1 p q. The inequality between power eans of orders q/p 1and1for positive nubers b i, i 1, 2,...,, states that bq/p i p/q b i, 2.4 whence for any fixed j 1, 2,...n, after substitution of b i, i 1, 2,...,weobtain a q 1j aq 2j j p/q aq p/q 1 a p 1j ap 2j ap j, 2.5

Journal of Inequalities and Applications 5 whence after suation over j we find that n a q p/q n 1j aq 2j aq j p/q 1 p/q 1 a p i. 2.6 Because p 1, the inequality between power eans of orders p and 1 iplies that p a p i 1 p a i. 2.7 The above inequality and 2.6 iediately yield 1 1/q p/q. 2.8 Case 2 1 q<p. If n, then C 1 in 1.4, and a related proof is the sae as that for the following case when 0 <q 1 p. Now suppose that >n. By the inequality for power eans of orders p/q 1and1, we obtain n a q 1j aq 2j aq j n p/q q/p n a q 1j aq 2j aq j n n a q i1 aq i2 aq in. 2.9 Next, by the inequality for power eans of orders q 1and1, weobtain a q i1 aq i2 aq in a q i1 aq i2 aq in 1/q q. 2.10 For any fixed i {1, 2,...,} the inequality 2.1 of Lea 2.1 with s p>q r iplies that a q 1/q. i1 aq i2 aq in a p i1 ap i2 in ap 2.11

6 Journal of Inequalities and Applications Obviously, inequalities 2.9, 2.10, and 2.11 iediately yield n 1 q/p q 1 p/q q/p q, 2.12 which is actually inequality 1.4 with the constant C n 1/q 1 1/q. Case 3 0 <q 1 p. By inequality 2.1 with r q and s p, for each j 1, 2,...,n,we obtain a q p/q 1j aq 2j aq p j a 1j ap 2j ap j, 2.13 whence after suation over j, we have n a q 1j aq 2j aq j p/q n a p i1 ap i2 ap in a p i. 2.14 By the inequality for power eans of orders p 1and1, weget ap i a i 2.15 or equivalently a p i 1 a i 1. 2.16 The above inequality and 2.14 iediately yield 1 p/q, 2.17 as desired. Case 4 0 <q p<1. The proof can be obtained fro those of Case 1, by replacing with in each related inequality.

Journal of Inequalities and Applications 7 Case 5 0 <p<q<1. If n, then the proof is the sae as that for Case 6. If>n, then the proof can be obtained fro those of Case 2, by replacing with in each related inequality. Case 6 0 <p<1 q. For any fixed j 1, 2,...,n, inequality 2.1 of Lea 2.1 with r q and s p gives a q p/q 1j aq 2j aq p j a 1j ap 2j ap j, 2.18 whence after suation over j,weget n a q p/q n 1j aq 2j aq j a p i. 2.19 As > 1, for positive integers b 1,b 2,...,b, there holds b i p b i, 2.20 whence for any fixed j 1, 2,...n, after substitution of b i a p i, i 1, 2,...,weobtain a p i 1 a i 1. 2.21 The above inequality and 2.19 iediately yield 1 p/q, 2.22 and the proof is copleted. 3. The Integral Analogue of Theore 1.1 Let X, Σ,μ be a easure space with a positive Borel easure μ. For any 0 <p< let L p L p μ denote the usual Lebesgue space consisting of all μ-easurable coplex-valued functions f : X C such that X f p dμ <. 3.1

8 Journal of Inequalities and Applications Recall that the usual nor p of f L p is defined as f p X f p dμ if p 1; f p X f p dμ if 0 <p<1. The following result is the integral analogue of Theore 1.1. Theore 3.1. For given 0 <p< let u 1,u 2,...,u be arbitrary functions in L p. Then, if 1 p<, we have u 1 p u p 1 in{1/2,} u 1 2 u 2 p. 3.2 If 0 <p<1, then u 1 p u p 1 u 1 2 u 2 p. 3.3 Both inequalities are sharp For 1 <p 2 the equality in 3.2 and 3.3 is attained if u 1 u 2 u a.e. on X. Ifp>2 or 0 <p<1, then the equality is attained for u i χ Ei, where E i are μ-easurable sets with i 1, 2,...,, such that μ E 1 μ E 2 μ E n and E i E j whenever i / j. Proof. The proof of each inequality is copletely siilar to the corresponding one given in Theore 1.1 with a fixed q 2. For clarity, we give here only a proof related to the case when 1 p 2. Applying the inequality between power eans of orders 2/p 1and1tothe functions u i p i 1,...,, we have Integrating the above relation, we obtain which can be written in the for X p/2 u i 2 p/2 1 u i. p 3.4 p/2 u i 2 dμ p/2 1 u i p dμ, 3.5 X p u 1 2 u 2 1/2 u i p dμ X u i p p 3.6 u i p. Obviously, the above inequality yields 3.2 for 1 <p 2.

Journal of Inequalities and Applications 9 Corollary 3.2. Let p 1, and let w u iv be a coplex function in L p. Then there holds the sharp inequality u p v p 2 1 in 1/2, u iv p. 3.7 References 1 H. Minkowski, Geoetrie der Zahlen, Teubner, Leipzig, Gerany, 1910. 2 G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cabridge Univerity Press, Cabridge, UK, 1952. 3 E. F. Beckenbach and R. Bellan, Inequalities, vol. 30 of Ergebnisse der Matheatik und ihrer Grenzgebiete, Springer, Berlin, Gerany, 1961. 4 H. Tôyaa, On the inequality of Ingha and Jessen, Proceedings of the Japan Acadey, vol. 24, no. 9, pp. 10 12, 1948. 5 H. Alzer and S. Ruscheweyh, A converse of Minkowski s inequality, Discrete Matheatics, vol. 216, no. 1 3, pp. 253 256, 2000.