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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 646034, 14 pages doi:10.1155/2010/646034 Research Article Improvement and Reversion of Slater’s Inequality and Related Results M. Adil Khan 1 and J. E. Pe ˇ cari ´ c 1, 2 1 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan 2 Faculty of Textile Technology, University of Zagreb, Zagreb 10002, Croatia Correspondence should be addressed to M. Adil Khan, adilbandai@yahoo.com Received 6 March 2010; Accepted 2 June 2010 Academic Editor: Kunquan Lan Copyright q 2010 M. Adil Khan and J. E. Pe ˇ cari ´ c. 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. We use an inequality given by Mati ´ candPe ˇ cari ´ c 2000 and obtain improvement and reverse of Slater’s and related inequalities. 1. Introduction In 1981 Slater has proved an interesting companion inequality to Jensen’s inequality 1. Theorem 1.1. Suppose that φ : I ⊆ R → R is increasing convex function on interval I,for x 1 ,x 2 , ,x n ∈ I ◦ (where I ◦ is the interior of the interval I) and for p 1 ,p 2 , ,p n ≥ 0 with P n n i1 p i > 0,if n i1 p i φ x i > 0,then 1 P n n i1 p i φ x i ≤ φ n i1 p i φ x i x i n i1 p i φ x i . 1.1 When φ is strictly convex on I, inequality 1.1 becomes equality if and only if x i c for some c ∈ I ◦ and for all i with p i > 0. It was noted in 2 that by using the same proof the following generalization of Slater’s inequality 1981 can be given. 2 Journal of Inequalities and Applications Theorem 1.2. Suppose that φ : I ⊆ R → R is convex function on interval I,forx 1 ,x 2 , ,x n ∈ I ◦ (where I ◦ is the interior of the interval I) and for p 1 ,p 2 , ,p n ≥ 0 with P n n i1 p i > 0.Let n i1 p i φ x i / 0, n i1 p i φ x i x i n i1 p i φ x i ∈ I ◦ , 1.2 then inequality 1.1 holds. When φ is strictly convex on I, inequality 1.1 becomes equality if and only if x i c for some c ∈ I ◦ and for all i with p i > 0. Remark 1.3. For multidimensional version of Theorem 1.2 see 3. Another companion inequality to Jensen’s inequality is a converse proved by Dragomir and Goh in 4. Theorem 1.4. Let φ : I ⊆ R → R be differentiable convex function defined on interval I.Ifx i ∈ I,i 1, 2, ,n n ≥ 2 are arbitrary members and p i ≥ 0 i 1, 2, ,n with P n n i1 p i > 0, and let x 1 P n n i1 p i x i , y 1 P n n i1 p i φ x i . 1.3 Then the inequalities 0 ≤ y − φ x ≤ 1 P n n i1 p i φ x i x i − x 1.4 hold. In the case when φ is strictly convex, one has equalities in 1.4 if and only if there is some c ∈ I such that x i c holds for all i with p i > 0. Mati ´ candPe ˇ cari ´ cin5 proved more general inequality from which 1.1 and 1.4 can be obtained as special cases. Theorem 1.5. Let φ : I ⊆ R → R be differentiable convex function defined on interval I and let x i ,p i ,P n , x, and y be stated as in Theorem 1.4.Ifd ∈ I is arbitrary chosen number, then one has y ≤ φ d 1 P n n i1 p i x i − d φ x i . 1.5 Also, when φ is strictly convex, one has equality in 1.5 if and only if x i d holds for all i with p i > 0. Remark 1.6. If φ, x i ,p i ,P n , and x are stated as in Theorem 1.4 and we let n i1 p i φ x i / 0, also if x n i1 p i x i φ x i / n i1 p i φ x i ∈ I, then by setting d x in 1.5, we get Slater’s inequality 1.1 and similarly by setting d x in 1.5,weget1.4. Journal of Inequalities and Applications 3 The following refinement of 1.4 is also valid 5. Theorem 1.7. Let φ : I ⊆ R → R be strictly convex differentiable function defined on interval I and let x i ,p i ,P n , x, and y be stated as in Theorem 1.4 and d φ −1 1/P n n i1 p i φ x i , then the inequalities y ≤ φ d 1 P n n i1 p i φ x i x i − d , 1.6 0 ≤ y − φ x ≤ φ d 1 P n n i1 p i φ x i x i − d − φ x ≤ 1 P n n i1 p i φ x i x i − x 1.7 hold. The equalities hold in 1.6 and in 1.7 if and only if x 1 x 2 ··· x n . Remark 1.8. In 6 Dragomir has also proved Theorem 1.7. In this paper, we use an inequality given in 5 and derive two mean value theorems, exponential convexity, log-convexity, and Cauchy means. As applications, such results are also deduce for related inequality. We use some log-convexity criterion and prove improvement and reverse of Slater’s and related inequalities. We also prove some determinantal inequalities. 2. Mean Va lue Theorems Theorem 2.1. Let φ ∈ C 2 I,whereI is closed interval in R, and let P n n i1 p i , p i > 0, x i ,d ∈ I with x i / d i 1, 2, ,n and y 1/P n n i1 p i φx i . Then there exists ξ ∈ I such that φ d 1 P n n i1 p i x i − d φ x i − y φ ξ 2P n n i1 p i x i − d 2 . 2.1 Proof. Since φ x is continuous on I, m ≤ φ x ≤ M for x ∈ I, where m min x∈I φ x and M max x∈I φ x. Consider the functions φ 1 , φ 2 defined as φ 1 x Mx 2 2 − φ x , φ 2 x φ x − mx 2 2 . 2.2 Since φ 1 x M − φ x ≥ 0, φ 2 x φ x − m ≥ 0, 2.3 φ i x for i 1, 2 are convex. 4 Journal of Inequalities and Applications Now by applying φ 1 for φ in inequality 1.5, we have Md 2 2 − φ d 1 P n n i1 p i x i − d Mx i − φ x i − 1 P n n i1 p i Mx 2 i 2 − φ x i ≥ 0. 2.4 From 2.4 we get φ d 1 P n n i1 p i x i − d φ x i − y ≤ M 2P n n i1 p i x i − d 2 , 2.5 and similarly by applying φ 2 for φ in 1.5,weget φ d 1 P n n i1 p i x i − d φ x i − y ≥ m 2P n n i1 p i x i − d 2 . 2.6 Since n i1 p i x i − d 2 > 0asx i / d, p i > 0 i 1, 2, ,n , 2.7 by combining 2.5 and 2.6, we have m ≤ 2P n φ d 1/P n n i1 p i x i − d φ x i − y n i1 p i x i − d 2 ≤ M. 2.8 Now using the fact that for m ≤ ρ ≤ M there exists ξ ∈ I such that φ ξρ,weget2.1. Corollary 2.2. Let φ ∈ C 2 I,whereI is closed interval in R, and let x i , x, y, and P n be stated as in Theorem 1.4 with p i > 0 and x i / x i 1, 2, ,n. Then there exists ξ ∈ I such that φ x 1 P n n i1 p i x i − x φ x i − y φ ξ 2P n n i1 p i x i − x 2 . 2.9 Proof. By setting d x in Theorem 2.1,weget2.9. Theorem 2.3. Let φ, ψ ∈ C 2 I,whereI is closed interval in R, and let P n n i1 p i , p i > 0 and x i ,d ∈ I with x i / d i 1, 2, ,n. Then there exists ξ ∈ I such that φ ξ ψ ξ φ d 1/P n n i1 p i x i − d φ x i − 1/P n n i1 p i φ x i ψ d 1/P n n i1 p i x i − d ψ x i − 1/P n n i1 p i ψ x i , 2.10 provided that the denominators are nonzero. Journal of Inequalities and Applications 5 Proof. Let the function k ∈ C 2 I be defined by k c 1 φ − c 2 ψ, 2.11 where c 1 and c 2 are defined as c 1 ψ d 1 P n n i1 p i x i − d ψ x i − 1 P n n i1 p i ψ x i , c 2 φ d 1 P n n i1 p i x i − d φ x i − 1 P n n i1 p i φ x i . 2.12 Then, using Theorem 2.1 with φ k, we have 0 c 1 φ ξ 2P n − c 2 ψ ξ 2P n n i1 p i x i − d 2 , 2.13 because kd1/P n n i1 p i x i − dk d − 1/P n n i1 p i kx i 0. Since 1/P n n i1 p i x i − d 2 > 0asx i / d and p i > 0 i 1, 2, ,n, therefore, 2.13 gives us c 2 c 1 φ ξ ψ ξ . 2.14 After putting the values of c 1 and c 2 ,weget2.10 . Corollary 2.4. Let φ, ψ ∈ C 2 I,whereI is closed interval in R, and P n n i1 p i , p i > 0 and let x i ∈ I, x 1/P n n i1 p i x i with x i / x i 1, 2, ,n. Then there exists ξ ∈ I such that φ ξ ψ ξ φ x 1/P n n i1 p i x i − x φ x i − 1/P n n i1 p i φ x i ψ x 1/P n n i1 p i x i − x ψ x i − 1/P n n i1 p i ψ x i , 2.15 provided that the denominators are nonzero. Proof. By setting d x in Theorem 2.3,weget2.15. Corollary 2.5. Let x i ,d ∈ I with x i / d and P n n i1 p i , p i > 0 i 1, 2, ,n. Then for u, v ∈ R \{0, 1}, u / v,thereexistsξ ∈ I,whereI is positive closed interval, such that ξ u−v v v − 1 d u u/P n n i1 p i x i − d x u−1 i − 1/P n n i1 p i x u i u u − 1 d v v/P n n i1 p i x i − d x v−1 i − 1/P n n i1 p i x v i . 2.16 Proof. By setting φxx u and ψxx v , x ∈ I,inTheorem 2.3,weget2.16 . 6 Journal of Inequalities and Applications Corollary 2.6. Let x i ∈ I, P n n i1 p i , p i > 0 i 1, 2, ,n, and x 1/P n n i1 p i x i with x i / x. Then for u, v ∈ R \{0, 1}, u / v,thereexistsξ ∈ I,whereI is positive closed interval, such that ξ u−v v v − 1 x u u/P n n i1 p i x i − x x u−1 i − 1/P n n i1 p i x u i u u − 1 x v v/P n n i1 p i x i − x x v−1 i − 1/P n n i1 p i x v i . 2.17 Proof. By setting φxx u and ψxx v , x ∈ I,in2.15,weget2.17. Remark 2.7. Note that we can consider the interval I m x ,M x , where m x min i {x i ,d}, M x max i {x i ,d}. Since the function ξ → ξ u−v with u / v is invertible, then from 2.16 we have m x ≤ v v − 1 d u u/P n n i1 p i x i − d x u−1 i − 1/P n n i1 p i x u i u u − 1 d v v/P n n i1 p i x i − d x v−1 i − 1/P n n i1 p i x v i 1/u−v ≤ M x . 2.18 We will say that the expression in the middle is a mean of x i ,d. From 2.17 we have min i { x i } ≤ v v − 1 x u u/P n n i1 x i − x x u−1 i − 1/P n n i1 p i x u i u u − 1 x v v/P n n i1 p i x i − x x v−1 i − 1/P n n i1 p i x v i 1/u−v ≤ max i { x i } . 2.19 The expression in the middle of 2.19 is a mean of x i . In fact similar results can also be given for 2.10 and 2.15. Namely, suppose that φ /ψ has inverse function, then from 2.10 and 2.15 we have ξ φ ψ −1 φ d 1/P n n i1 p i x i − d φ x i − 1/P n n i1 p i φ x i ψ d 1/P n n i1 p i x i − d ψ x i − 1/P n n i1 p i ψ x i . ξ φ ψ −1 φ x 1/P n n i1 p i x i − x φ x i − 1/P n n i1 p i φ x i ψ x 1/P n n i1 p i x i − x ψ x i − 1/P n n i1 p i ψ x i . 2.20 So, we have that the expression on the right-hand side of 2.20 is also means. 3. Improvements and Related Results Definition 3.1 see 7, page 2.Afunctionφ : I → R is convex if φ s 1 s 3 − s 2 φ s 2 s 1 − s 3 φ s 3 s 2 − s 1 ≥ 0 3.1 holds for every s 1 <s 2 <s 3 , s 1 ,s 2 ,s 3 ∈ I. Journal of Inequalities and Applications 7 Lemma 3.2 see 8. Let one define the function ϕ t x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x t t t − 1 ,t / 0, 1, − log x, t 0, x log x, t 1. 3.2 Then ϕ t xx t−2 , that is, ϕ t is convex for x>0. Definition 3.3 see 9.Afunctionφ : I → R is exponentially convex if it is continuous and n k,l1 a k a l φ x k x l ≥ 0, 3.3 for all n ∈ N, a k ∈ R, and x k ∈ I, k 1, 2, ,nsuch that x k x l ∈ I, 1 ≤ k, l ≤ n, or equivalently n k,l1 a k a l φ x k x l 2 ≥ 0. 3.4 Corollary 3.4 see 9. If φ is exponentially convex function, then det φ x k x l 2 n k,l1 ≥ 0 3.5 for every n ∈ N x k ∈ I, k 1, 2, ,n. Corollary 3.5 see 9. If φ : I → 0, ∞ is exponentially convex function, then φ is a log-convex function that is φ λx 1 − λ y ≤ φ λ x φ 1−λ y , ∀x, y ∈ I, λ ∈ 0, 1 . 3.6 Theorem 3.6. Let x i ,p i ,d ∈ R i 1, 2, ,n, P n n i1 p i . Consider Γ t to be defined by Γ t ϕ t d 1 P n n i1 p i x i − d ϕ t x i − 1 P n n i1 p i ϕ t x i . 3.7 Then i for every m ∈ N and for every s k ∈ R,k∈{1, 2, 3, ,m}, the matrix Γ s k s l /2 m k,l1 is a positive semidefinite matrix; particularly det Γ s k s l /2 m k,l1 ≥ 0; 3.8 ii the function t → Γ t is exponentially convex; 8 Journal of Inequalities and Applications iii if Γ t > 0, then the function t → Γ t is log-convex, that is, for −∞ <r<s<t<∞, one has Γ s t−r ≤ Γ r t−s Γ t s−r . 3.9 Proof. i Let us consider the function defined by μ x m k,l1 a k a l ϕ s kl x , 3.10 where s kl s k s l /2,a k ∈ R for all k ∈{1, 2, 3, ,m},x>0 Then we have μ x m k,l1 a k a l x s kl −2 m k1 a k x s k −2/2 2 ≥ 0. 3.11 Therefore, μx is convex function for x>0. Using μx in inequality 1.5,weget m k,l1 a k a l Γ s kl ≥ 0, 3.12 so the matrix Γ s k s l /2 m k,l1 is positive semi-definite. ii Since lim t → 0 Γ t Γ 0 and lim t → 1 Γ t Γ 1 ,soΓ t is continuous for all t ∈ R,x>0, and we have exponentially convexity of the function t → Γ t . iii Let Γ t > 0, then by Corollary 3.5 we have that Γ t is log-convex, that is, t → log Γ t is convex, and by 3.1 for −∞ <r<s<t<∞ and taking φtlog Γ t ,weget t − s log Γ r r − t log Γ s s − r log Γ t ≥ 0, 3.13 which is equivalent to 3.9. Corollary 3.7. Let x i ,p i ∈ R i 1, 2, ,n, P n n i1 p i and x 1/P n n i1 p i x i . Consider Γ t to be defined by Γ t ϕ t x 1 P n n i1 p i x i − x ϕ t x i − 1 P n n i1 p i ϕ t x i . 3.14 Then i for every m ∈ N and for every s k ∈ R,k∈{1, 2, 3, ,m}, the matrix Γ s k s l /2 m k,l1 is a positive semi-definite matrix. Particularly det Γ s k s l /2 m k,l1 ≥ 0, 3.15 Journal of Inequalities and Applications 9 ii the function t → Γ t is exponentially convex; iii if Γ t > 0, then the function t → Γ t is log-convex, that is, for −∞ <r<s<t<∞, one has Γ s t−r ≤ Γ r t−s Γ t s−r . 3.16 Proof. To get the required results, set d x in Theorem 3.6. Let x x 1 ,x 2 , ,x n be positive n-tuple and p 1 ,p 2 , ,p n positive real numbers, and let P n n i1 p i .LetM t x denote the power mean of order t t ∈ R, defined by M t x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 P n n i1 p i x t i 1/t ,t / 0, n i1 x p i i 1/P n ,t 0. 3.17 Let us note t hat M 1 xx. By 2.18 we can give the following definition of Cauchy means. Let x i ,d ∈ I with x i / d, I is positive closed interval, and P n n i1 p i , p i > 0 i 1, 2, ,n, M u,v Γ u Γ v 1/u−v 3.18 for −∞ <u / v<∞ are means of x i ,d. Moreover we can extend these means to the other cases. So by limit we have M u,u exp P n d u log d u − 1 n i1 p i x u i log x i P n M u u x − d u n i1 p i x u−1 i log x i P n M u−1 u−1 x P n d u u − 1 M u u x − duM u−1 u−1 x − 2u − 1 u u − 1 ,u / 0, 1, M 0,0 exp P n log 2 d − P n M 2 2 log x 2P n log M 0 x − 2d n i1 p i x −1 i log x i 2P n log d − log M 0 x 1 − dM −1 −1 x 1 , M 1,1 exp P n d log 2 d 2 n i1 p i x i log x i − dP n M 2 2 log x − 2logM 0 x 2 P n d log d − 1 P n x − dP n log M 0 x − 1 , 3.19 where log x log x 1 , log x 2 , ,log x n . 10 Journal of Inequalities and Applications Theorem 3.8. Let t, s, u, v ∈ R such that t ≤ u, s ≤ v, then the following inequality is valid: M t,s ≤ M u,v . 3.20 Proof. For convex function φ it holds that 7, page 2 φ x 2 − φ x 1 x 2 − x 1 ≤ φ y 2 − φ y 1 y 2 − y 1 3.21 with x 1 ≤ y 1 , x 2 ≤ y 2 , x 1 / x 2 , y 1 / y 2 . Since by Theorem 3.6, Γ t is log-convex, we can set in 3.21: φxlog Γ x , x 1 t, x 2 s, y 1 u,andy 2 v, then we get log Γ s − log Γ t s − t ≤ log Γ v − log Γ u v − u . 3.22 From 3.22 we get 3.20 for s / t and u / v. For s t and u v we have limiting case. Similarly by 2.19 we can give the following definition of Cauchy type means. Let x i ∈ I with x i / x, I is positive closed interval, and P n n i1 p i ,p i > 0 i 1, 2, ,n, M u,v Γ u Γ v 1/u−v 3.23 for −∞ <u / v<∞ are means of x i . Moreover we can extend these means to the other cases. So by limit we have M u,u exp P n x u log x u − 1 n i1 p i x u i log x i P n M u u x − x u n i1 p i x u−1 i log x i P n M u−1 u−1 x P n x u u − 1 M u u x − xuM u−1 u−1 x − 2u − 1 u u − 1 ,u / 0, 1, M 0,0 exp P n log 2 x − P n M 2 2 log x 2P n log M 0 x − 2 x n i1 p i x −1 i log x i 2P n log x − log M 0 x 1 − xM −1 −1 x 1 , M 1,1 exp P n x log 2 x 2 n i1 p i x i log x i − xP n M 2 2 log x 2logM 0 x 2 P n x log x − 1 P n x − xP n log M 0 x − 1 , 3.24 where log x log x 1 , log x 2 , ,log x n . [...]... special case of s1 s2 /2 if s1 < s2 and for t s1 , r s2 , and s s1 s2 /2 if 3.39 for t s1 , s s2 , r s2 < s1 Similarly by setting m 2 in 3.44 , we have special case of 3.40 for r s1 , s s2 , s1 s2 /2 if s2 < s1 and t s1 s2 /2 if s1 < s2 and for r s2 , s s1 , and t Acknowledgments The research of the first and second authors was funded by Higher Education Commission, Pakistan The research of the second... generalization of Slater’s inequality, ” Journal of Approximation Theory, c c vol 44, no 3, pp 292–294, 1985 4 S S Dragomir and C J Goh, “A counterpart of Jensen’s discrete inequality for differentiable convex mappings and applications in information theory,” Mathematical and Computer Modelling, vol 24, no 2, pp 1–11, 1996 5 M Mati´ and J E Peˇ ari´ , “Some companion inequalities to Jensen’s inequality, ”... Ministry of Science, Education, and Sports under the Research Grant 117-1170889-0888 References 1 M L Slater, “A companion inequality to Jensen’s inequality, ” Journal of Approximation Theory, vol 32, no 2, pp 160–166, 1981 2 J E Peˇ ari´ , “A companion to Jensen-Steffensen’s inequality, ” Journal of Approximation Theory, vol 44, c c no 3, pp 289–291, 1985 3 J E Peˇ ari´ , “A multidimensional generalization of. .. defined by 3.30 Proof By setting d ds1 and d d s1 s2 /2 in Theorem 3.6 i , we get the required results Remark 3.12 We note that H t, t Ft So by setting m 2 in 3.35 , we have special case of s1 s2 /2 if s1 < s2 and for t s1 , r s2 , and s s1 s2 /2 3.28 for t s1 , s s2 , and r if s2 < s1 Similarly by setting m 2 in 3.36 , we have special case of 3.29 for r s1 , s s1 s2 /2 if s1 < s2 and for r s2 , s...Journal of Inequalities and Applications 11 Theorem 3.9 Let t, s, u, v ∈ R such that t ≤ u, s ≤ v, then the following inequality is valid: Mt,s ≤ Mu,v 3.25 Proof The proof is similar to the proof of Theorem 3.8 Let Mt x be stated as above, define dt as dt ⎧ t ⎪ Mt x , ⎪ ⎪ t−1 ⎪M x ⎪ ⎪ t−1 ⎪ ⎨ M x, ⎪ −1... S Dragomir, “On a converse of Jensen’s inequality, ” Univerzitet u Beogradu Publikacije Elektrotehniˇ kog c Fakulteta Serija Matematika, vol 12, pp 48–51, 2001 7 J E Peˇ ari´ , F Proschan, and Y L Tong, Convex Functions, Partial Orderings, and Statistical Applications, c c vol 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992 8 M Anwar and J E Peˇ ari´ , “On logarithmic... Anwar and J E Peˇ ari´ , “On logarithmic convexity for differences of power means and related c c results,” Journal of Mathematical Inequalities, vol 4, no 2, pp 81–90, 2009 9 M Anwar, J E Jakˇ eti´ , J Peˇ ari´ , and Atiq ur Rehman, “Exponential convexity, positive semi-definite s c c c matrices and fundamental inequalities,” Journal of Mathematical Inequalities, vol 4, no 2, pp 171–189, 2010 ... have special case of 3.29 for r s1 , s s1 s2 /2 if s1 < s2 and for r s2 , s s1 , t s1 s2 /2 if s2 < s1 s2 , t Journal of Inequalities and Applications 13 Let Mt x be stated as above, define dt as dt ϕt −1 1 n pi ϕt xi Pn i 1 Mt−1 x , t ∈ R 3.37 The following improvement and reverse of inequality 1.6 are also valid Theorem 3.13 Let xi , pi , dt ∈ R for all i Gt ϕt dt 1 Pn n i 1 1, 2, , n, Pn n pi xi... pi xi ϕt xi n i 1 pi ϕt xi t / 0, 1, t 0, t 3.26 1 The following improvement and reverse of Slater’s inequality are valid Theorem 3.10 Let xi , pi , dt ∈ R 1, 2, , n , Pn i Ft ϕt dt − 1 Pn n i 1 pi Let Ft be defined by n pi ϕt xi 3.27 i 1 Then i Ft ≥ H s; t t−r / s−r H r; t s−t / s−r , 3.28 s−t / s−r , 3.29 for −∞ < r < s < t < ∞ and −∞ < t < r < s < ∞ ii Ft ≤ H s; t t−r / s−r H r; t for −∞ < r... Inequalities and Applications n Theorem 3.14 Let xi , pi , dt ∈ R i 1, 2, , n , Pn i 1 pi Then for every m ∈ N and for every sk ∈ R, k ∈ {1, 2, 3, , m}, the matrices K sk sl /2, s1 m 1 , K sk sl /2, s1 s2 /2 m 1 are positive semi-definite matrices Particularly k,l k,l det K det K sl sk 2 sk 2 m , s1 sl s1 , k,l 1 s2 2 ≥ 0, m k,l 1 ≥ 0, 3.43 3.44 where K s, t is defined by 3.41 Proof By setting d ds1 and . Corporation Journal of Inequalities and Applications Volume 2010, Article ID 646034, 14 pages doi:10.1155/2010/646034 Research Article Improvement and Reversion of Slater’s Inequality and Related Results M log-convexity, and Cauchy means. As applications, such results are also deduce for related inequality. We use some log-convexity criterion and prove improvement and reverse of Slater’s and related inequalities distribution, and reproduction in any medium, provided the original work is properly cited. We use an inequality given by Mati ´ candPe ˇ cari ´ c 2000 and obtain improvement and reverse of Slater’s and
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