New Inequalities For Convex Sequences With Applications

Transcription

1 It. J. Ope Problems Comput. Math., Vol. 5, No. 3, September, 0 ISSN ; Copyright c ICSRS Publicatio, 0 New Iequalities For Covex Sequeces With Applicatios Zielaâbidie Latreuch ad Beharrat Belaïdi Departmet of Mathematics, Laboratory of Pure ad Applied Mathematics, Uiversity of Mostagaem UMAB, B. P. 7 Mostagaem, Algeria. z.latreuch@gmail.com ad belaidi@uiv-mosta.dz. Abstract I this paper, we will show some ew iequalities for covex sequeces, ad we will also make a coectio betwee them ad Chebyshev s iequality, which implies the existece of ew class of sequeces satisfyig Chebyshev s iequality. We give also some applicatios ad geeralizatio of Haber ad Mercer s iequalities. Keywords: Chebyshev s iequality, Covex Sequeces, Symmetric sequeces. Itroductio ad mai results A classic result due to Chebyshev see [, 5, 6, 0,, 3] is stated i the followig theorem. Theorem A Let a = a, a,, a ad b = b, b,, b be two sequeces of real umbers mootoic i the same directio, ad p = p, p,, p be a positive sequece. The p i a i b i p i b i.. p i p i a i If a ad b are mootoic i opposite directios, the the reverse of the iequality i. holds. I either case equality holds if ad oly if either a = a = = a or b = b = = b.

2 6 Zielaâbidie Latreuch ad Beharrat Belaïdi There exist several results which show that Chebyshev iequality is valid uder weaker coditios, for example the coditio that the sequeces be mootoic ca be replaced by the coditio that they be similarly ordered. I this case Theorem A is a simple cosequece of the followig idetity p i a i b i p i b i p i = j= p i a i p i p j a i a j b i b j.. Note that the sequeces a = a, a,, a ad b = b, b,, b are said to be similarly ordered if a i a j b i b j 0, i, j.3 holds, ad they are said to be oppositely ordered if the reverse iequality holds. Cosiderable attetio has bee give to the study of covex sequeces ad their properties, ad the correspodig iequalities with applicatios. I geeral, covex sequeces as discrete versios of covex fuctios play a importat role i mathematical aalysis ad i the theory of iequalities. Iequalities for covex sequeces provided cosiderable iterest i provig a large umber of elegat results with applicatios see Wu ad Shi [5], Wu ad Debath [6] ad Mercer [9]. I additio, several authors icludig Mitriovic ad Vasic [], Roberts ad Varberg [4], ad Mitriovic et al. [0] preseted a large umber of major results for covex sequeces ad related iequalities. The aim of this paper is to prove ew type of iequalities for covex sequeces, ad we put a lik betwee these iequalities ad Chebyshev s iequality. Before we state our results we give the followig defiitio. Defiitio A [7] Let a = a, a,, a be a sequece of real umbers, a is a covex sequece if for all i =,,, we have a i + a i+ a i+. If the above iequality reversed, the a is termed cocave sequece. We obtai the followig results.

3 New Iequalities For Covex Sequeces With Applicatios 7 Theorem. Let a = a, a,, a ad b = b, b,, b be two covex cocave sequeces, ad p = p, p,, p be a positive sequece symmetric about [ ] pk = p k, for all k =,,. The p i a i b i + p i a i b i p i a i p i b i..4 p i If a is covex or cocave ad b is cocave or covex sequeces, the the iequality.4 is reversed. I either case equality holds if ad oly if either a = a = = a or b = b = = b. Corollary. Let a = a, a,, a ad b = b, b,, b be two covex cocave sequeces. If either a or b is symmetric about [ ], the a i b i a i b i..5 If a is covex or cocave ad b is cocave or covex sequeces, the the iequality.5 is reversed. I either case equality holds if ad oly if either a = a = = a or b = b = = b. Theorem. Let a = a, a,, a ad b = b, b,, b be two covex or cocave sequeces. i If a ad b are similarly ordered, the a i b i a i b i + a i b i a i b i..6 ii If a ad b are oppositely ordered, the a i b i a i b i a i b i..7 Theorem.3 Let a = a, a,, a ad b = b, b,, b be two sequeces of real umbers where a is covex sequece ad b decreasig for all k =,, [ ] [ ad icreasig for all k = ],,. The the iequality.4 holds.

4 8 Zielaâbidie Latreuch ad Beharrat Belaïdi Here we obtai the discrete versio of Fejér [3] double iequality. Theorem.4 Let a = a, a,, a be a covex sequece of real umbers ad p = p, p,, p be a positive sequece symmetric about [ ]. The a N + a N a + a p i p i a i p i..8 If a = a, a,, a is cocave sequece the the iequality.8 is reversed. Some lemmas Lemma. Let a = a, a,, a be covex or cocave sequece of real umbers. The the sequece c = c, c,, c, where c k = a k + a k. is decreasig icreasig for all k =,, [ ] ad icreasig decreasig for all k = [ ],,. Proof. Suppose that a is covex sequece. Sice c is a symmetric sequece about [ ], the we eed oly to prove that c is decreasig for all k =,, [ ]. We have c k c k+ = a k + a k a k+ + a k = a k + a k+ a k+ + + a k a k + a k a k+ + a k+ a k+ + + a k a k + a k = a k + a k+ a k+ + a k+ + a k+3 a k+ + + a k + a k a k. for all k =,, [ ]. By usig mathematical iductio ad., we obtai c k c k+ = k i=k a i + a i+ a i If a is a cocave sequece, the by usig similar proof we obtai the result. Lemma. Let a = a, a,, a ad b = b, b,, b be two sequeces of real umbers. If a ad b are similarly ordered, the a i b i a i b i..4

5 New Iequalities For Covex Sequeces With Applicatios 9 If a ad b are oppositely ordered, the the iequality.4 is reversed. Proof. Sice a ad b are similarly ordered, the we have for all i =,, which implies that The It follows that a i a i b i b i 0.5 a i b i + a i b i a i b i + a i b i..6 a i b i = a i b i + a i b i a i b i + a i b i = a i b i..7 a i b i a i b i. If a ad b are oppositely ordered, the by usig similar proof we obtai the result. I the followig we deote N { c i = where c = c, c,, c ad N = [ ]. c + c + + c N, if is eve, c + c + + c N + c N, if is odd, Lemma.3 Let a = a, a,, a ad b = b, b,, b be two sequeces of real umbers ad p = p, p,, p be a positive sequece, we deote by N = [ ]. If a ad b are similarly ordered, the N N N N p i p i a i b i p i a i p i b i..8 If a ad b are oppositely ordered, the the iequality.8 is reversed. Proof.i If is eve, the the iequality.8 is equivalet to N N N N p i p i a i b i p i a i p i b i

6 0 Zielaâbidie Latreuch ad Beharrat Belaïdi which is Chebychev s iequality. ii If is odd, we have c i = c + c + + c N. Sice a ad b are similarly ordered, the a i a j b i b j 0, i, j which implies a i b i + a j b j a i b j + a j b i, i, j..9 Multiplyig both sides of iequality.9 by p i p j which implies p i p j a i b i + p i p j a j b j p i p j a i b j + p i p j a j b i, i, j.0 p j p a b + p p j a j b j p a p j b j + p b p j a j, p j p a b + p p j a j b j p a p j b j + p b p j a j, p j p N a N b N + p N p j a j b j p N a N p j b j + p N b N p j a j, p jp N a N b N + p Np j a j b j p Na N p j b j + p Nb N p j a j,. for all j N. Summig both sides of iequalities. with respect to i =,, N, we obtai p j p i a i b i + p j a j b j p i p j b j p i a i + p j a j p i b i.. By the same reasoig as before we have by usig. N j= N j= N p j p i a i b i + N p j b j p i a i + N j= N j= N p j a j b j p i N p j a j p i b i.3 which is equivalet to.8. If a ad b are oppositely ordered, the by usig similar proof we obtai the result.

7 New Iequalities For Covex Sequeces With Applicatios 3 Proof of the Theorems Proof of Theorem. Without loss of geerality we suppose that a ad b are covex sequeces ad we deote by U ad V the followig sequeces U i = a i + a i, V i = b i + b i. Sice a ad b are covex sequeces, the by usig Lemma. we deduce that U ad V have the same directio of mootoy. By applyig Lemma.3 for all i =,, N = [ ], we obtai N N N N p i p i U i V i p i U i p i V i, 3. where p = p, p,, p is a positive sequece ad symmetric about [ ]. The p i a i b i + a i b i + p i a i b i + a i b i N p i Usig the idetities N N p i a i + a i p i b i + b i. 3. p i a i b i + a i b i = p i a i b i, 3.3 ad p i a i b i + a i b i = p i a i b i, 3.4 p i = p i, 3.5 p i a i + a i = p i a i, p i b i + b i = p i b i. 3.6 By usig , we obtai from 3. p i a i b i + p i a i b i p i a i p i b i. p i

8 Zielaâbidie Latreuch ad Beharrat Belaïdi Now, if a covex cocave ad b cocave covex sequeces, the by usig similar proof as above we obtai the result. Proof of Theorem. i Sice a ad b are covex sequeces ad similarly ordered, the by Lemma. we have a i b i a i b i 3.7 which we ca write a i b i a i b i + a i b i. 3.8 By Theorem. ad 3.8, we have a i b i a i b i + a i b i a i b i. 3.9 ii Sice a ad b are covex sequeces, the by Theorem. a i b i a i b i a i b i a i b i. 3.0 O the other had, we have a i b i a i b i, 3. because a ad b are oppositely ordered. By 3.0 ad 3., we get a i b i a i b i. 3. Now, if a ad b are cocave sequeces, the by usig similar proof as above we obtai the result. Proof of Theorem.3 We deote by U ad V the followig sequeces U i = a i + a i, 3.3

9 New Iequalities For Covex Sequeces With Applicatios 3 V i = b i + b i. 3.4 Sice U is covex sequece, the by Lemma., U is decreasig for all i =,, [ ] [ ad icreasig for all i = ],,. I order to prove.4 we eed to prove that V is decreasig for all i =,, [ ] ad icreasig for all i = [ ] [,,. Let i ] ], we deote by j = + i j. The [ V i V i+ = b i + b i b i+ + b i = b i b i+ + b i b i = b i b i+ + b j b j because b is decreasig for all i =,, [ ] [ ad icreasig for all i = ],,. By the same method we ca prove easily that V is icreasig for all i = [ ],,. The we have U ad V havig the same directio of mootoy, ad by applyig Theorem A with p = p, p,, p is a positive sequece symmetric about [ ], we obtai iequality.4. Proof of Theorem.4 Suppose that a = a, a,, a is a covex sequece. By applyig Lemma. for the sequece v k = a k + a k we obtai the followig iequalities v N v k v, for all k =,, N 3.6 ad v N v k v, for all k = N,,. 3.7 By 3.6 ad 3.7 we deduce that a N + a N a k + a k a + a, k =,,. 3.8 Multiplyig iequalities 3.8 by p k, we obtai for all k =,, a N + a N p k a k + a k p k a + a p k, k =,, which implies a N + a N p k k= p k a k a + a p k. k= k= For the case of cocave sequece we use similar proof.

10 4 Zielaâbidie Latreuch ad Beharrat Belaïdi 4 Some Applicatios I 978, S. Haber [4] proved the followig iequality: Theorem B Let a ad b be o egative real umbers, the for every 0, we have a + a b + + b a + b. + May authors are iterested by this iequality see [, 4, 8]. It s easy to show that x k = a k b k a 0, b 0 k = 0,,, is a covex sequece. The by Theorem.4, we have for x k = a k b k a 0, b 0 k = 0,,, ad p =,,, hece + k=0 x k x 0 + x a + a b + + b a + b + which is the upper boud of Haber iequality, ad we ca state: Theorem 4. Let a ad b be o egative real umbers, the for every 0, we have a + b a + a b + + b a + b. + A. McD. Mercer geeralized Haber iequality for covex sequeces ad obtaied the followig result:, Theorem C [8] Let {u} be covex sequece of real umbers. The + u i Cu i i. I this sectio we prove that Mercer iequality ca be deduced by Theorem.3. It s clear that the symmetric sequece about [ ] v i = + Ci

11 New Iequalities For Covex Sequeces With Applicatios 5 is decreasig for i = 0,, [ ] [ ad icreasig for i = ],,. The by applyig Theorem.3 for the sequeces u i, v i i = 0,, where v i is a covex sequece ad p =,,, we obtai ad sice v i = 0, we obtai u i v i + + u i u i v i Cu i i 0. Theorem 4. Let a i i N be a covex ad symmetric sequece of real umbers such that a i > 0. The the polyomial do t have ay o egative zero. P x = a x + a x + + a x + a 0 Proof. Suppose that x 0. It s clear that b i = x i i =,,, is a covex sequece for x 0. The by applyig.5 we obtai for x P x = a i x i + a i, x i x = a i > 0. + x For x = the result is trivial. This completes the proof of Theorem 4.. Remark 4. Puttig p =,,, i Theorem.4, it s clear that we have equality i Theorem.4 if ad oly if that a = a,, a is arithmetic sequece i. e., a i+ + a i = a i+ for all i =,, ad.8 become a i = a + a, which is the sum of terms of arithmetic sequece.

12 6 Zielaâbidie Latreuch ad Beharrat Belaïdi 5 Ope problem I 950, M. Bieracki, H. Pidek ad C. Ryll-Nardjewski [0, Chapter X] established the followig discrete versio of Grüss iequality: Theorem D Let a = a, a,, a, b = b, b,, b be two -tuples of real umbers such that r a i R ad s b i S for i =,,,. The oe has a i b i a i b i [ ] [ ] R r S s, where [x] deotes the iteger part of x, x R. The followig questio arises: Ca we obtai a aalogue result for covex -tuples a = a, a,, a ad b = b, b,, b? Refereces [] H. Alzer ad J. E. Pe carić, O a iequality of A.M. Mercer, Rad Hrvatske Akad. Za. Umjet. No , [] P. L. Ceby sev, Poloe Sobraie Sočieiĭ. Russia [Complete Collected Works] Izdatelstvo Akademii Nauk SSSR, Moscow-Leigrad.] 946, 947, 948, vol., 34 pp.; vol., 50 pp.; vol. 3, 44 pp.; vol. 4, 55 pp. [3] L. Fejér, Über die Fourierreihe, II., Math. Naturwiss Az. Ugar. Akad. Wiss. Hugaria, vol. 4, 906, [4] S. Haber, A elemetary iequality, Iterat. J. Math. Math. Sci., 979, o 3, [5] G. H. Hardy, J. E. Littlwood ad G. Pólya, Iequalities, d ed. Cambridge, at the Uiversity Press, 95. [6] S. J. Karli ad W. J. Studde, Tchebycheff systems: With applicatios i aalysis ad statistics, New York: Itersciece Publishers, 966. [7] V. I. Levi ad S. B. Ste cki, Iequalities, Amer. Math. Soc. Trasl , 9. [8] A. McD. Mercer, A ote o a paper: A elemetary iequality by S. Haber, Iterat. J. Math. Math. Sci., 6 983, o 3,

13 New Iequalities For Covex Sequeces With Applicatios 7 [9] A. McD. Mercer, Polyomials ad covex sequece iequalities. J. Iequal. Pure Appl. Math. 6005, o., Art. 8, 4 pp. [0] D. S. Mitriović, J. E. Pe carić ad A. M. Fik, Classical ad ew iequalities i aalysis. Mathematics ad its Applicatios East Europea Series, 6. Kluwer Academic Publishers Group, Dordrecht, 993. [] D. S. Mitriović ad P. M. Vasić, History, variatios ad geeralisatios of the Čebyšev iequality ad the questio of some priorities, Publ. Elektroteh. Fak. Uiv. Beogr. Ser. Mat. Fiz. No , 30. [] D. S. Mitriović, Aalytic Iequalities, I Cooperatio with P. M. Vasić, Spriger-Verlag, New York-Berli, 970. [3] J. E. Pe carić, F. Proscha ad Y. L. Tog, Covex fuctios, partial orderigs, ad statistical applicatios, Mathematics i Sciece ad Egieerig. 87. Academic Press Ic. Bosto, MA, 99. [4] A.W. Roberts ad D.E. Varberg, Covex Fuctios, Pure ad Applied Mathematics, Vol. 57, Academic Press, New York-Lodo, 973. [5] S. Wu ad H. Shi, Majorizatio proofs of iequalities for covex sequeces, Math. Practice Theory 33003, o., [6] S. Wu ad L. Debath, Iequalities for covex sequeces ad their applicatios, Comput. Math. Appl., , o. 4,