Research Article Refinements of the Lower Bounds of the Jensen Functional

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1 Abstract and Applied Analysis Volume 20, Article ID 92439, 3 pages doi:0.55/20/92439 Research Article Refinements of the Lower Bounds of the Jensen Functional Iva Franjić, Sadia Khalid, 2 and Josip Pečarić 2, 3 Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 0000 Zagreb, Croatia 2 Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahe 54600, Pakistan 3 Faculty of Textile Technology, University of Zagreb, Prilaz Baruna Filipovića 28A, 0000 Zagreb, Croatia Crespondence should be addressed to Sadia Khalid, saadiakhalid76@gmail.com Received July 20; Accepted 4 August 20 Academic Edit: Wing-Sum Cheung Copyright q 20 Iva Franjić et al. 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 iginal wk is properly cited. The lower bounds of the functional defined as the difference of the right-hand and the left-hand side of the Jensen inequality are studied. Refinements of some previously known results are given by applying results from the they of majization. Furtherme, some interesting special cases are considered.. Introduction The classical Jensen inequality states see e.g.,. Theem. see 2. Let I be an interval in R, and let f : I R be a convex function. Let n 2, x x,...,x n I n, and let p p,...,p n be a positive n-tuple, that is, such that p i > 0 f i,...,n,then f P n i ) p i x i p i f x i, P n i. where P n n i p i.iff is strictly convex, then inequality. is strict unless x x n.

2 2 Abstract and Applied Analysis In this wk, the functional J x, p,f ) P n p i f x i f i P n i ) p i x i.2 defined as the difference of the right-hand and the left-hand sides of the Jensen inequality is studied. Me precisely, its lower bounds are investigated, together with various sets of assumptions under which they hold. The lower bounds of J x, p,f were the topic of interest in many papers. F example, the following results were proved in 3 see also, page 77. In what follows, I is an interval in R. Theem.2. Let f : I R be a convex function, x I n, and let p be a positive n-tuple, then P n J x, p,f ) max j k n { p j f x j ) pk f x k p j p k ) f pj x j p k x k p j p k )} 0..3 Theem.3. Let f : I R be a convex function and x I n.letp and r be positive n-tuples such that p r, that is, p i r i,i,...,n,then P n J x, p,f ) R n J x, r,f ) 0,.4 where P n n i p i and R n n i r i. Further, in 4, the following theem was given. An alternative proof of the same result was given in 5. Theem.4. Let f : I R be a convex function, n 2, and x I n.letpand q be positive n-tuples such that n i p i n i q i, then { pj max j n q j } J x, q,f ) J x, p,f ) min j n { pj q j } J x, q,f ) 0..5 F me related results, see 6 8. The motivation f the research in this wk were the following results presented in 9. Lemma.5. Let f be a convex function on I, p a positive n-tuple such that P n n i p i and x,x 2,...,x n I, n 3 such that x x 2 x n. F fixed x j,x j,...,x n,where j 2, 3,...,n, the Jensen functional J x, p,f defined in.2 is minimal when x x 2 x j x j, that is, J x, p,f ) P j f x j ) p i f x i f P j x j i j p i x i,.6 i j

3 Abstract and Applied Analysis 3 where P j p i, j,...,n..7 i Lemma.6. Let f be a convex function on I, p a positive n-tuple such that P n n i p i and x,x 2,...,x n I, n 3 such that x x 2 x n. F fixed x,x 2,...,x k,wherek 2, 3,...,n, the Jensen functional J x, p,f defined in.2 is minimal when x k x k x n x n, that is, where J x, p,f ) k k p i f x i f x k f p i x i x k ),.8 i i p i, k,...,n..9 Theem.7. Let f be a convex function on I, p a positive n-tuple such that P n n i p i and x,x 2,...,x n I, n 3 such that x x 2 x n. F fixed x j and x k,where j<k n, the Jensen functional J x, p,f defined in.2 is minimal when x x 2 x j, x k x k x n, x j x j 2 x k P jx j x k P j,.0 that is, J x, p,f ) P j f ) x j Qk f x k ) ) Pj x j x k P j f,. P j where P j are as in.7 and are as in.9. The key step in proving these results was the following lemma presented in the same paper. Lemma.8. Let f be a convex function on I, and let p,p 2 be nonnegative real numbers. If a,a 2,b,b 2 I are such that a,a 2 b,b 2 and p a p 2 a 2 p b p 2 b 2,.2 then p f a p 2 f a 2 p f b p 2 f b 2..3

4 4 Abstract and Applied Analysis Note that f a monotonic n-tuple x, Theem.7 is an improvement of Theem.2, in a sense that the maximum of the right-hand side of. is greater than the middle part of.3, which follows directly from the Jensen inequality. The aim of this wk is to give an improvement of Lemmas.5 and.6, and Theem.7, in a sense that the condition of monotonicity imposed on the n-tuple x will be relaxed. Several sets of conditions under which.6,.8,and. hold shall be given. In our proofs, in addition to Lemma.8,the following result from the they of majization is needed. It was obtained in 0. Lemma.9. Let f be a convex function on I, p a positive n-tuple, and a, b I n such that k p i a i i k p i b i f k, 2,...,n, i p i a i i p i b i. i.4 If a is a decreasing n-tuple, then one has p i f a i i p i f b i, i.5 while if b is an increasing n-tuple, then we have p i f b i i p i f a i. i.6 If f is strictly convex and a / b,then.5 and.6 are strict. Note that f n 2, inequality.5 holds if a 2 a b and if.2 is valid, while inequality.6 holds if a b b 2 and if.2 is valid. 2. Main Results In what follows, J x, p,f is as in.2, P j are as in.7, and,asin.9. Without any loss of generality, we assume that P n, since f positive n-tuples such that P n / results follow easily by substituting p i with p i /P n. Furtherme, f j<k n, we introduce the following notation: ) { } J min x, p,f min Pj, f )) ) xj x k x j f xk 2f, 2 ) J jk x, p,f Pj f ) k k x j p i f x i f x k f P j x j p i x i x k. i j i j 2. Note that J n x, p,f J x, p,f.

5 Abstract and Applied Analysis 5 Theem 2.. Let f be a convex function on I and p a positive n-tuple such that P n, n 2. Let j<k n and x i I, i,...,k.ifx j is such that P j i p i x i x j k Q j i j p i x i x k, 2.2 k Q j i j p i x i x k x j P j i 2.3 then one has J k x, p,f ) Jjk x, p,f ). 2.4 Proof. The claim is that ) k p i f x i f p i x i x k P j f ) x j f P j x j i i k p i x i x k. 2.5 i j As a simple consequence of the Jensen inequality., we have p i f x i P j f i P j i p i x i ). 2.6 Therefe, if we prove P j f P j i ) k p i x i f P j x j p i x i x k P j f ) k x j f p i x i x k ), 2.7 i j i the claim will follow. The idea is to apply Lemma.8 f p P j, p 2, a x j, a 2 k i p ix i x k, b /P j j i p ix i,andb 2 P j x j k i j p ix i x k. Condition.2 is obviously satisfied. In addition, we need to check that P j i P j i k p i x i x j P j x j p i x i x k, i j k k p i x i p i x i x k P j x j p i x i x k. i i j 2.8 Easy calculation shows that both of these conditions are valid if 2.2 holds. Thus, the claim follows from Lemma.8. Note that we could have taken p, p 2 P j, a k i p ix i x k,

6 6 Abstract and Applied Analysis a 2 x j, b P j x j k i j p ix i x k,andb 2 /P j j i p ix i, instead. In this case, the necessary conditions would follow from 2.3. Theem 2.2. Let the conditions of Theem 2. hold. If x j is such that P j i k p i x i x j p i x i x k, i k p i x i x k x j P j i i then inequality 2.4 holds. Proof. Proof is analogous to the proof of Theem 2.. Instead of Lemma.8, weapply Lemma.9 f n 2 and the same choice of weights and points, their obvious rearrangement. Theem 2.3. Let f be a convex function on I and p a positive n-tuple such that P n, n 2. Let j<k n and x i I, i j,...,n.ifx k is such that P k k P j x j p i x i x k i j p i x i x k P k 2. k P j x j p i x i, 2.2 i j then one has J jn x, p,f ) Jjk x, p,f ). 2.3 Proof. Similarly as in the proof of Theem 2., after first applying the Jensen inequality to the sum on the left-hand side, the claim will follow if we prove f ) k p i x i f P j x j p i x i x k x k f P j x j i j p i x i. i j 2.4 We can apply Lemma.8 f p, p 2, a P j x j n i j p ix i,a 2 x k,b P j x j k i j p ix i x k,andb 2 / n p ix i, since condition.2 is obviously satisfied and 2. ensures that the rest of the necessary conditions are fulfilled, and thus the claim is proved. After the obvious rearrangement, applying Lemma.8 with 2.2, the claim is recaptured.

7 Abstract and Applied Analysis 7 Theem 2.4. Let the conditions of Theem 2.3 hold. If x k is such that P j x j p i x i x k i j p i x i x k P j x j i j then inequality 2.3 holds. Proof. It is analogous to the proof of Theem 2.3. Instead of Lemma.8,we apply Lemma.9 f n 2 and the same choice of weights and points, their obvious rearrangement. Collary 2.5. Let f be a convex function on I and p a positive n-tuple such that P n, n 2. Let x I n be a real n-tuple and j<k n. If x k is such that P k i k p i x i x k p i x i x k P k k i and x j is such that either holds, then one has J x, p,f ) J k x, p,f ) Jjk x, p,f ). 2.9 If x j is such that P j i p i x i x j Q j i j Q j i j p i x i x j P j i and x k is such that either holds, then one has J x, p,f ) J jn x, p,f ) Jjk x, p,f ) Proof. The first inequality in 2.9 follows from Theem 2.3 f j, and the second is a direct consequence of Theem 2., while the first inequality in 2.22 follows from Theem 2. f k n, and the second is a consequence of Theem 2.3.

8 8 Abstract and Applied Analysis Collary 2.6. Let the conditions of Collary 2.5 hold. If x k is such that p i x i x k i p i x i x k i and x j is such that either holds, then inequality 2.9 holds. If x j is such that P j i p i x i x j i 2.25 p i x i x j P j i i 2.26 and x k is such that either holds, then inequality 2.22 holds. Proof. The first inequality in 2.9 follows from Theem 2.3 f j, and the second is a direct consequence of Theem 2., while the first inequality in 2.22 follows from Theem 2. f k n, and the second is a consequence of Theem 2.3. Theem 2.7. Let f be a convex function on I and p a positive n-tuple such that P n, n 2. Let x I n be a real n-tuple, and let j<k n.ifx j and x k are such that P j i p i x i x j p i x i x k i 2.27 p i x i x k p i x i x j P j i i 2.28 then one has J x, p,f ) J jk x, p,f ) Proof. The claim is that ) p i f x i p i f x i f p i x i i i P j f ) k x j Qk f x k f P j x j p i x i x k. i j 2.30

9 Abstract and Applied Analysis 9 After applying the Jensen inequality to the two sums on the left-hand side, we need to prove f ) k p i x i f P j x j p i x i x k P j f i j ) f x k f p i x i P j f ) x j. i P j i ) p i x i 2.3 Set p, p 2, p 3 P j, a x k,a 2 n i p ix i, a 3 x j,b / n p ix i,b 2 P j x j k i j p ix i x k,andb 3 /P j j i p ix i. Assumption 2.27 ensures that the necessary conditions of Lemma.9 f n 3 are fulfilled, and so 2.3 follows from.5. By obvious rearrangement, utilizing 2.28, the inequality is recaptured. Remark 2.8. Note that conditions 2.9 and 2.23 combined together give a condition P j i k p i x i x j p i x i x k i p i x i x k i 2.32 while 2.5 and 2.25 combined together give P j i p i x i x j p i x i P j x j p i x i x k i i j 2.33 both of which are me restricting than The same is true f combining conditions 2.0 and 2.24, 2.6 and 2.26, and comparing the result with Theem 2.9. Let f be a convex function on I and p a positive n-tuple such that P n, n 2. Let j<k n and x i I, i j,...,k, then one has ) J jk x, p,f Pj f ) x j Qk f x k ) ) Pj x j x k P j f P j J min x, p,f ) Proof. The first inequality is an immediate consequence of the Jensen inequality. The other two follow immediately from.4. Remark 2.0. Inequalities 2.9, 2.22,and 2.34 recapture results from Lemmas.5 and.6, and Theem.7 as special cases, since an increasing n-tuple x fulfils conditions 2.2 and 2.7, thatis, 2. and A decreasing n-tuple x, on the other hand, fulfills conditions 2.3 and 2.8, thatis, 2.2 and 2.2. The proofs of Theem 2.9 and Collary 2.5, that is, Theems 2. and 2.3, are in fact analogous to the proofs of Theem.7, Lemmas.5 and.6 from 9.

10 0 Abstract and Applied Analysis 3. Some Special Cases In this section, we consider some special cases of the presented results. The same special cases were considered in 9, but here we obtain them under me relaxed conditions on the n-tuple x. Me precisely, Collaries 2.5 and 2.6, Theem 2.7, after applying Theem 2.9, yield J x, p,f ) P j f ) x j Qk f x k ) ) Pj x j x k P j f. 3. P j Collary 3.. Let the conditions of Collaries 2.5 and 2.6, Theem 2.7 hold, then p i a i i n i a p i i P j a j a k ) P j / P j P j a j a / P j. k 3.2 Proof. This follows from 3. f f x e x, using notation a i e x i. Collary 3.2. Let the conditions of Collaries 2.5 and 2.6, Theem 2.7 hold, and let in addition x i > 0, i,...,n,then n i p ix i n i xp i i x P j j x k ) Pj Pj x j x k. 3.3 P j Proof. Follows from 3. f f x ln x. Collary 3.3. Let the conditions of Collaries 2.5 and 2.6, Theem 2.7 hold, and let in addition x i > 0, i,...,n,then i p i x i n i p P j xk x j ). ix i x j x k Pj x j x k 3.4 Proof. This follows from 3. f f x /x. In 9, additional bounds of J x, p,f, lower than those obtained in the previous collaries, were derived f the case f x e x and f x /x. Now, note that from Theem 2.9, under conditions of Collaries 2.5 and 2.6,Theem 2.7, we have J x, p,f ) J min x, p,f ) Next, we compare estimates obtained from 3.5 with those obtained in 9. Case. F f x e x, using notation a i e x i, inequality 3.5 takes the fm p i a i i n a p i i i min { P j, } ak a j. 3.6

11 Abstract and Applied Analysis In 9, under the assumption that a is an increasing n-tuple, the following inequality was obtained p i a i i n a p i i i C ak a j, 3.7 where 2P j, P j, C P j, P j. 3.8 Note that when P j, 3.6 recaptures this result. However, when P j, the constant C is better, since 2P j / P j P j. Case 2. F f x /x andx i > 0, i,...,n, inequality 3.5 takes the fm i p i x i n i p min { } x k x j P j, ). ix i x j x k xj x k 3.9 In 9, under the assumption that x is an increasing n-tuple such that x > 0, the following inequality was obtained: i p i x i n i p ix i C xk x j x j x k, 3.0 where P j, P j 3, C 4P j, P j P j In der to compare these two estimates, first assume that P j. Since P j xk x j x j x k xj x k ) P j xk x j x j x k xk x j xj x k, 3.2 it follows that the estimate in 3.9 is better than the one in 3.0. Next, assume that P j 2. First, observe that xk x j Pj xk x j ) Pj xk x j x k x j. 3.3

12 2 Abstract and Applied Analysis Simple calculation reveals that xk x j 2, 3.4 x k x j and so we conclude that the estimate in 3.9 is better than the one in 3.0 when P j x k x j 2 / x k x j, while when x k x j 2 / x k x j P j 2, the estimate in 3.0 isbetterthantheonein 3.9. Further, assume that 2 P j 3. In this case, the estimate in 3.0 is better than the one in 3.9, thatis, P j xk x j ) Qk xk x j. 3.5 Namely, P j xk x j ) 2Qk xk x j ), 2 xk x j ) Qk xk x j xk x j Finally, if 3 P j, the estimate in 3.0 is again better than the one in 3.9, thatis, 4P j ) xk xj x k Qk. x j P j 3.7 This is equivalent to P j 3x j 3x k 2 x j x k ) xk x j. 3.8 In this case, we have P j 3x j 3x k 2 x j x k ) 3x j 3x k 2 x j x k ), 3.9 and since 3x j 3x k 2 x j x k ) xk x j xk x j 0, 3.20 the claim follows. Acknowledgments This research wk was partially funded by the Higher Education Commission, Pakistan. The research of the auths was suppted by the Croatian Ministry of Science, Education and

13 Abstract and Applied Analysis 3 Spts, under the Research Grants nos first auth and third auth. References D. S. Mitrinović, J. E. Pečarić,andA.M.Fink,Classical and New Inequalities in Analysis, vol. 6 of Mathematics and Its Applications, Kluwer Academic Publishers, Ddrecht, The Netherlands, J. L. W. V. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes, Acta Mathematica, vol. 30, no., pp , S. S. Dragomir, J. Pečarić, and L. E. Persson, Properties of some functionals related to Jensen s inequality, Acta Mathematica Hungarica, vol. 70, no. -2, pp , S. S. Dragomir, Bounds f the nmalised Jensen functional, Bulletin of the Australian Mathematical Society, vol. 74, no. 3, pp , J. Barić, M. Matić, and J. E. Pečarić, On the bounds f the nmalized Jensen functional and Jensen- Steffensen inequality, Mathematical Inequalities & Applications, vol. 2, no. 2, pp , S. Abramovich, S. Ivelić, and J. E. Pečarić, Improvement of Jensen-Steffensen s inequality f superquadratic functions, Banach Mathematical Analysis, vol. 4, no., pp , S. Ivelić, A. Matković, and J. E. Pečarić, On a Jensen-Mercer operat inequality, Banach Mathematical Analysis, vol. 5, no., pp. 9 28, M. Khosravi, J. S. Aujla, S. S. Dragomir, and M. S. Moslehian, Refinements of Choi-Davis-Jensen s inequality, Bulletin of Mathematical Analysis and Applications, vol. 3, no. 2, pp , V. Cirtoaje, The best lower bound depended on two fixed variables f Jensen s inequality with dered variables, Inequalities and Applications, vol. 200, Article ID 28258, 2 pages, N. Latif, J. Pečarić, and I. Perić, On majization of vects,favard and Berwald inequalities, In press.

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