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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 720615, 15 pages doi:10.1155/2010/720615 Research Article Generalization of Stolarsky Type Means J. Peˇcari´c 1, 2 and G. Roqia 2 1 Faculty of Textile Technology, University of Zagreb, Pierottijeva, 6, 10000 Zagreb, Cr oatia 2 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan Correspondence should be addressed to G. Roqia, rukiyya@gmail.com Received 27 April 2010; Revised 10 August 2010; Accepted 15 October 2010 Academic Editor: Paolo E. Ricci Copyright q 2010 J. Pe ˇ cari ´ c and G. Roqia. 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 generalize means of Stolarsky type and show the monotonicity of these generalized means. 1. Introduction and Preliminaries The following double inequality is well known in the literature as the Hermite-Hadamard H.H integral inequality f a b 2 ≤ 1 b − a b a f x dx ≤ f a f b 2 , 1.1 provided that f : a, b → is a convex function 1, page 137, 2,page1. This result for convex functions plays an important role in nonlinear analysis. These classical inequalities have been improved and generalized in a number of ways and applied for special means including Stolarsky type, logarithmic, and p-logarithmic means. A generalization of H.H inequalities was obtained in 3–5, 2,page5,and1, page 143. Theorem 1.1. Let p, q be positive real numbers and a 1 , a, b, b 1 be real numbers such that a 1 ≤ a< b ≤ b 1 . Then the inequalities f pa qb p q ≤ 1 2y Ay A−y f x dx ≤ pf a qf b p q 1.2 2 Journal of Inequalities and Applications hold for A pa qb/p q, y>0, and all continuous convex functions f : a 1 ,b 1 → if and only if y ≤ b − a / p q min p, q . 1.3 Remark 1.2. The inequalities given by 1.2 are strict if f is a continuous strictly convex on a 1 ,b 1 . If we keep the assumptions as stated in Theorem 1.1,wealsohave1, page 146 1 2y Ay A−y f x dx −f pa qb p q ≤ pf a qf b p q − 1 2y Ay A−y f x dx. 1.4 The above inequality is strict, when f is strictly convex continuous function. Let us define F i : Ca, b → for i 1, 2, 3bydifferences of 1.2 and 1.4 F 1 f; p, q; a, b, y pf a qf b p q − 1 2y M m f x dx, F 2 f; p, q; a, b, y 1 2y M m f x dx −f pa qb p q , F 3 f; p, q; a, b, y pf a qf b p q f pa qb p q − 1 y M m f x dx, 1.5 where m A −y, M A y. Remark 1.3. It is clear from inequalities 1.2 and 1.4 that if the conditions of Theorem 1.1 are satisfied and f ∈ K 2 a, bf is continuous convex on a, b,then F i f; p, q; a, b, y ≥ 0, for i 1, 2, 3. 1.6 Consider the following means: E r,t x, y ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ r y t − x t t y r − x r 1/t−r ,tr t − r / 0, y r − x r r log y −log x 1/r ,r / 0,t 0, e −1/r x x r y y r 1/x r −y r ,t r / 0, √ xy, t r 0, 1.7 Journal of Inequalities and Applications 3 where x, y ∈ 0, ∞ such that x / y and r, t ∈ . These means are known as Stolarsky means. Namely, Stolarsky introduced these means in 1975 see 1, page 120 and proved that for r ≤ u and t ≤ v one can get E r,t x, y ≤ E u,v x, y for x, y ∈ 0, ∞ ,x / y. 1.8 Some simple proofs of inequality 1.8 and related results on means of Stolarsky type are given in 6. The aim of this paper is to prove the exponential convexity of the functions deduced from 1.5 and apply these functions to generalize the means of Stolarsky type, and at last we prove the monotonicity property of these new means. We review some necessary definitions and preliminary results. Definition 1.4 see 7.Afunctionf : a, b → is exponentially convex if it is continuous and n i,j1 ξ i ξ j f x i x j ≥ 0, 1.9 for each n ∈ N and every ξ i ∈ , i 1, ,nsuch that x i x j ∈ a, b,1≤ i, j ≤ n. Proposition 1.5 see 7. Let f : a, b → , be a function. Then f is exponentially convex if and only if f is continuous and n i,j1 ξ i ξ j f x i x j 2 ≥ 0, 1.10 for all n ∈ N, ξ i ∈ and x i ∈ a, b, 1 ≤ i ≤ n. Definition 1.6 see 1.Afunctionf : I → ,whereI is an interval in ,issaidtobelog- convex if log f is convex, or equivalently if for all x, y ∈ I and all α ∈ 0, 1, one has f αx 1 −α y ≤ f α x f 1−α y . 1.11 Corollary 1.7 see 7. If f : a, b → is exponentially convex then f is log -convex function. 4 Journal of Inequalities and Applications The following lemma is another way to define convex function 1,page2. Lemma 1.8. If f is a convex on an interval I ⊆ ,then f s 1 s 3 − s 2 f s 2 s 1 − s 3 f s 3 s 1 − s 2 ≥ 0 1.12 holds for each s 1 <s 2 <s 3 ,wheres 1 ,s 2 ,s 3 ∈ I. In Section 2, we prove the exponential and logarithmic convexity of t he functions deduced from 1.5. We also prove related mean value theorems of Cauchy type. 2. Main Results The following lemma gives us very important family of convex functions. Lemma 2.1 see 7. Consider a family of functions φ r : 0, ∞ → , r ∈ defined as φ r x ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ x r r r −1 ,r / 0, 1, −log x, r 0, x log x, r 1. 2.1 Then φ r is convex on 0, ∞ for each r ∈ . Theorem 2.2. Let p, q, a, b, A,andy be positive real numbers such that a<b, A pa qb p q ,y≤ b − a p q min p, q , i r : F i φ r ; p, q; a, b, y ,i 1, 2, 3, 2.2 where φ r is defined in Lemma 2.1.Then i matrix i r j r k /2 n j,k1 is positive semidefinite for each n ∈ N and r 1 , ,r n ∈ ; particularly, det i r j r k 2 p j,k1 ≥ 0for1≤ p ≤ n; 2.3 ii the function r → i r is exponentially convex on ; Journal of Inequalities and Applications 5 iii if i r > 0, then the function r → i r is a log-convex on and the following inequality holds for r, s, t ∈ such that r<s<t; i s t−r ≤ i r t−s i t s−r . 2.4 Proof. i Consider the function μ x n j,k1 u j u k φ r jk x 2.5 for 1 ≤ p ≤ n, x>0u j ∈ ,whereu j is not identically zero and r jk r j r k /2 μ x n j,k1 u j u k x r jk −2 ⎛ ⎝ n j1 u j x r j /2−1 ⎞ ⎠ 2 ≥ 0 ,x>0. 2.6 This shows that μ is a convex function for x>0. By setting f μ in 1.5, respectively and from Remark 1.3,weget n j,k1 u j u k pφ r jk a qφ r jk b p q − 1 2y M m φ r jk x dx ≥ 0, n j,k1 u j u k 1 2y M m φ r jk x dx −φ r jk pa qb p q ≥ 0, n j,k1 u j u k pφ r jk a qφ r jk b p q φ r jk pa qb p q − 1 y M m φ r jk x dx ≥ 0, 2.7 or equivalently n j,k1 u j u k i r jk ≥ 0. 2.8 6 Journal of Inequalities and Applications Therefore the given matrix is a positive semidefinite. By using well-known Sylvester criterion, we have det i r j r k 2 p j,k1 ≥ 0foreach1≤ p ≤ n. 2.9 ii Since lim r →l i r i l for l 0, 1, it follows that i is continuous on . Therefore, by Proposition 1.5 for f i , we get exponential convexity of i on . iii Let i r > 0, then the log-convexity of i is a simple consequence of Corollary 1.7. By setting f log i , s 1 r, s 2 s, s 3 t in Lemma 1.8,wehave t − r log i s ≤ t − s log i r s − r i t , 2.10 which implies 2.4. We will use the fo llowing lemma in the proof of mean value theorem. Lemma 2.3 see 1,page4. Let f ∈ C 2 a, b such that α ≤ f x ≤ β ∀x ∈ a, b . 2.11 If one considers the functions h 1 , h 2 ,definedby h 1 x αx 2 2 − f x , h 2 x f x − βx 2 2 , 2.12 then h 1 and h 2 are convex on a, b. Proof. Therefore h 1 x α − f x ≥ 0, h 2 x f x − β ≥ 0, 2.13 that is, h j for j 1, 2areconvexona, b. Theorem 2.4. Let p, q, a, b, A,andy be real numbers as given in Theorem 1.1.Iff ∈ C 2 a, b then there exists ξ ∈ a, b such that F i f; p, q; a, b, y f ξ 2 F i x 2 ; p, q; a, b,y for i 1, 2, 3. 2.14 Journal of Inequalities and Applications 7 Proof. Since f ∈ C 2 a, b, we can take that α ≤ f ≤ β.NowinRemark1.3,replacingf by h j , j 1, 2 defined in Lemma 2.3,wehave F i h j ; p, q; a, b, y ≥ 0forj 1, 2. 2.15 This gives F i f x ; p, q : a, b, y ≤ β 2 F i x 2 ; p, q; a, b, y , α 2 F i x 2 ; p, q; a, b, y ≤ F i f x ; p, q; a, b, y . 2.16 Combining 2.16 and 14,weget α 2 F i x 2 ; p, q; a, b, y ≤ F i f x ; p, q; a, b, y ≤ β 2 F i x 2 ; p, q; a, b, y . 2.17 By using Remark 1.2 F i x 2 ; p, q; a, b, y > 0, 2.18 therefore α ≤ 2F i f x ; p, q; a, b, y F i x 2 ; p, q; a, b, y ≤ β. 2.19 We get the required result. Theorem 2.5. Let p, q, a, b, A,andy be real numbers as given in Theorem 1.1.Iff, g ∈ C 2 a, b such that g x do not vanish for any x ∈ a, b, then there exits ξ ∈ a, b such that F i f; p, q; a, b, y F i g; p, q; a, b, y f ξ g ξ for i 1, 2, 3. 2.20 Proof. Define functions φ i ∈ C 2 a, b, i 1, 2, 3by φ i c i 1 g −c i 2 f, 2.21 8 Journal of Inequalities and Applications where c i 1 F i f; p, q; a, b, y , c i 2 F i g; p, q; a, b, y . 2.22 Then using Theorem 2.4 for f φ i ,wehave 0 c i 1 g ξ 2 − c i 2 f ξ 2 F i x 2 ; p, q; a, b, y . 2.23 Using Remark 1.2 F i x 2 ; p, q; a, b, y > 0, 2.24 therefore c i 1 c i 2 f ξ g ξ , 2.25 which is clearly 2.20. Corollary 2.6. If p, q, a, b, A,andy are real numbers as defined in Theorem 1.1 then for −∞ <r, t<∞, r / t, r / 0, 1 and there exists ξ ∈ a, b such that ξ r−t t t − 1 F i x r ; p, q; a, b, y r r −1 F i x t ; p, q; a, b, y for i 1, 2, 3. 2.26 Remark 2.7. If the inverse of f /g exists, then from 2.20 we get ξ f g −1 F i f; p, q; a, b, y F i g; p, q; a, b, y for i 1, 2, 3. 2.27 Journal of Inequalities and Applications 9 3. Means of Stolarsky Type Expression 2.27 gives the means. We can consider E i r,t p, q; a, b, y F i φ r ; p, q; a, b, y F i φ t ; p, q; a, b, y 1/r−t ,r / t, for i 1, 2, 3 3.1 as a means in the broader sense. Moreover we can extend these means in other cases. Consider the following functions to cover all continuous e xtensions of 3.1: 1 r ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 r r −1 pa r qb r p q − M r1 − m r1 2y r 1 ,r / − 1, 0, 1, pb qa 2ab p q − log M − log m 4y ,r −1, M log M − 1 − m log m − 1 2y − p log a q log b p q ,r 0, pa log a qb log b p q − Γ,r 1, 2 r ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 r r −1 M r1 − m r1 2y r 1 − pa qb p q r ,r / −1, 0, 1, log M −log m 4y − p q 2 pa qb ,r −1, log pa qb p q − M log M −1 − m log m − 1 2y ,r 0, Γ − pa qb p q log pa qb p q ,r 1, 3 r ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 r r −1 pa r qb r p q pa qb p q r − M r1 − m r1 y r 1 ,r / −1, 0, 1, pb qa 2ab p q p q 2 pa qb − log M − log m 2y ,r −1, M log M − 1 − m log m − 1 y − p log a q log b p q − log pa qb p q ,r 0, pa log a qb log b p q pa qb p q log pa qb p q − 1 − 2Γ,r 1, 3.2 where ΓM 2 2logM − 1 − m 2 2logm − 1/8y. 10 Journal of Inequalities and Applications We have E i r,t p, q; a, b, y ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ F i φ r ; p, q; a, b, y F i φ t ; p, q;a,b,y 1/r−t ,r / t, exp 1 −2r r r −1 − F i φ 0 φ r ; p, q; a, b, y F i φ r ; p, q; a, b, y ,r t / − 1, 0, 1, exp 3 2 − F i φ 0 φ −1 ; p, q; a, b, y F i φ −1 ; p, q; a, b, y ,r t −1, exp 1 − F i φ 2 0 ; p, q; a, b, y 2F i φ 0 ; p, q; a, b, y ,r t 0, exp −1 − F i φ 0 φ 1 ; p, q; a, b, y 2F i φ 1 ; p, q; a, b, y ,r t 1, 3.3 for i 1, 2, 3. We will use t he following lemma to pr ove the monotonicity of Stolarsky type means. Lemma 3.1. Let f be log-convex function, and if r ≤ u, t ≤ v, r / t, u / v, then the following inequality is valid: f r f t 1/r−t ≤ f u f v 1/u−v . 3.4 The proof of this Lemma is given in 1. Theorem 3.2. Let p, q, a, b, A,andy be real numbers as defined in Theorem 1.1 and let r, t, u, v ∈ such that r ≤ u, t ≤ v, then the following inequality is valid: E i r,t p, q; a, b, y ≤ E i u,v p, q; a, b, y for i 1, 2, 3. 3.5 Proof. For a convex function φ, a simple consequence of the definition of convex function is the following inequality 1,page2: φ x 1 − φ x 2 x 2 − x 1 ≤ φ y 2 − φ y 1 y 2 − y 1 , with x 1 ≤ y 1 ,x 2 ≤ y 2 ,x 1 / x 2 ,y 1 / y 2 . 3.6 As i is log-convex we set φrlog i r, x 1 r, x 2 t, y 1 v, y 2 u in the above inequality and get log i r − log i t r − t ≤ log i u − log i v u −v , 3.7 which is equivalent to 3.5 for t / r, u / v.Bycontinuityof i , 3.5 is valid for t r, u v. [...]... completes the proof Remark 4.3 If we substitute p q 1, s → s − 1, and t → t − 1 in the above results, then the results of generalized Stolarsky type means proved in 6 are recaptured Acknowledgment This research was partially funded by Higher Education Commission, Pakistan The research of the rst author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant 117-1170889-0888...Journal of Inequalities and Applications 11 i Remark 3.3 If we substitute p q 1 and replace r → r − 1 and t → t − 1 in Er,t p, q; a, b, y , for i 1, 2, 3, then means of Stolarsky type and related results given in 6 are obtained 4 Generalized Means of Stolarsky Type By substiting a → as , b → bs , y → ys , r → r/s, t → t/s, ξ → ξ1/s in... Theorem 4.1 Theorem 2.2 is still valid if one sets φr ψr Proof The proof is similar to the proof of Theorem 2.2 Theorem 4.2 Let p, q, a, b, A, and y are real numbers as defined in Theorem 1.1 also let r, t, u, v ∈ Ê such that r ≤ u, t ≤ v, then the following inequality is valid: i i Er,t;s p, q; as , bs , ys ≤ Eu,v;s p, q; as , bs , ys Proof For s / 0, in this case we use Lemma 3.1 for f Ái r Ái t 1/... ys s / 0, t / r for i 1, 2, 3 4.2 To get all continuous extension of 4.2 , we consider ⎧ ⎪ pas qbs 1/s ⎨ , s / 0, A p q ⎪ p q 1/ p q ⎩ a b , s 0, ⎧ 1/s ⎪ bs − as ⎪ ⎪ min p, q , s / 0, ⎨ p q y≤ ⎪ b 1/ p q min{p,q} ⎪ ⎪ , s 0 ⎩ a 4.3 For s / 0, we define Áis r F i φr/s ; p, q; as , bs , ys for i 1, 2, 3, 4.4 where {φr ; r ∈ Ê} is the family of functions defined in Lemma 2.1 Here we have F i f; p, q; as ,... ⎪ ⎪ p q ⎪ ⎪ ⎪ ⎪ ⎪ pa log a qb log b ⎪ pas qbs s pas qbs ⎪ ⎪s ⎩ s log p q p q p q r 0, r s, p 2 pas q qbs − Ms 2 2 log Ms − 1 − ms 2 2 log ms − 1 /8ys − 2Γs , Journal of Inequalities and Applications For s 13 0, we consider a family of convex functions {ψr : r ∈ Ê} defined on ⎧ ⎪ 1 erx , ⎨ 2 r ⎪1 2 ⎩ x , 2 ψr x We have F i f; p, q; log a, log b, log y , i 1 2 log y F 2 f; p, q; log a, log b, log y F... I B Lackovi´ , “On an inequality for convex functions,” Univerzitet u Beogradu c c Publikacije Elektrotehniˇ kog Fakulteta Serija Matematika i Fizika, no 461–497, pp 63–66, 1974 c 4 A Lupas, “A generalization of Hadamard inequalities for convex functions,” Univerzitet u Beogradu ¸ Publikacije Elektrotehniˇ kog Fakulteta Serija Matematika i Fizika, no 544–576, pp 115–121, 1976 c 5 P M Vasi´ and I B Lackovi´... 544–576, pp 59–62, 1976 6 J Jakˇ eti´ , J Peˇ ari´ , and A ur Rehman, “On Stolarsky and related means,” Mathematical Inequalities s c c c and Applications, vol 13, pp 899–909, 2010 7 M Anwar, J Jakˇ eti´ , J Peˇ ari´ , and A 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 ... r2 ⎪1 ⎪ ⎩ log2 y, r 0, 6 ⎧ ⎪ 1 par qbr ap bq r/ p q y2r − 1 ⎪ ⎪ ap bq r/ p q − , ⎪ 2 ⎨r p q ryr log y ⎪ 1 p log2 a q log2 b 2 ⎪ ⎪ − log2 ap bq 1/ p q − log2 y , ⎪ ⎩2 p q 3 0, 4.9 r / 0, r 0 14 Journal of Inequalities and Applications We get means i Er,t;s s s p, q; a , b , y s ⎧ ⎪ Fi ⎪ ⎪ ⎪ ⎪ ⎪ Fi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪exp ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪exp ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪exp ⎪ ⎪ ⎨ 1/ r−t φr/s ; p, q; as... Science, Education and Sports under the Research Grant 117-1170889-0888 References 1 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 2 S S Dragomir and C E M Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monograph Collections,... as , bs , ys where i 1, 2, 3, ms f pas p As − ys , and Ms pf as p 1 2ys Ms f x dx − f ms qbs q As qf bs 1 − s q 2y pf as p ys Ms f x dx, ms pas p qbs , q qf bs 1 − s q y 4.5 Ms f x dx, ms 12 Journal of Inequalities and Applications We have Á1 s Á2 s Á3 s r r r where Γs ⎧ r s /s r s /s par qbr s Ms − ms ⎪ s2 ⎪ ⎪ , − ⎪r r − s ⎪ p q r s 2ys ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ pbs qas ⎪ log Ms − log ms ⎪ ⎪ ⎪ − , ⎪ ⎨ 2as bs p . Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 720615, 15 pages doi:10.1155/2010/720615 Research Article Generalization of Stolarsky Type Means J. Peˇcari´c 1,. 1.8 Some simple proofs of inequality 1.8 and related results on means of Stolarsky type are given in 6. The aim of this paper is to prove the exponential convexity of the functions deduced from. 1inE i r,t p, q; a, b, y, for i 1, 2, 3, then means of Stolarsky type and related results given in 6 are obtained. 4. Generalized Means of Stolarsky Type By substiting a → a s , b → b s , y → y s ,
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