Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID pptx

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 350973, 19 pages doi:10.1155/2011/350973 Research Article Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means ´ ´ ´ ˇ ´ Laszlo Horvath,1 Khuram Ali Khan,2, and J Pecaric2, Department of Mathematics, University of Pannonia, University Street 10, 8200 Veszpr´ m, Hungary e Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore 54600, Pakistan Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia Correspondence should be addressed to Khuram Ali Khan, khuramsms@gmail.com Received 26 November 2010; Accepted 23 February 2011 Academic Editor: Michel Chipot Copyright q 2011 L´ szlo Horv´ th et al This is an open access article distributed under the a ´ a Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited A recent refinement of the classical discrete Jensen inequality is given by Horv´ th and Peˇ ari´ a c c In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined We investigate the exponential convexity of some functions, study mean value theorems, and prove the monotonicity of the introduced means Introduction and Preliminary Results A new refinement of the discrete Jensen inequality is given in The following notations are also introduced in Let X be a set, P X its power set, and |X| denotes the number of elements in X Let u ≥ and v ≥ be fixed integers Define the functions Sv,w : {1, , u}v −→ {1, , u}v−1 , ≤ w ≤ v, Sv : {1, , u}v −→ P {1, , u}v−1 , Tv : P {1, , u}v −→ P {1, , u}v−1 1.1 Journal of Inequalities and Applications by Sv,w i1 , , iv : i1 , , iw−1 , iw , , iv , v S v i , , iv ≤ w ≤ v, {Sv,w i1 , , iv }, w ⎧ ⎪ ⎨ Tv I 1.2 S v i , , iv , I / φ, i1 , ,iv ∈I ⎪ ⎩φ, I φ Further, introduce the function αv,i : {1, , u}v −→ N, ≤ i ≤ u, 1.3 via αv,i i1 , , iv : Number of occurrences of i in the sequence i1 , , iv 1.4 For each I ∈ P {1, , u}v , let αI,i : αv,i i1 , , iv , ≤ i ≤ u 1.5 i1 , ,iv ∈I It is easy to observe from the construction of the functions Sv , Sv,w , Tv and αv,i that they not depend essentially on u, so we can write for short Sv for Su , and so on v H1 The following considerations concern a subset Ik of {1, , n}k satisfying αIk ,i ≥ 1, ≤ i ≤ n, 1.6 where n ≥ and k ≥ are fixed integers Next, we proceed inductively to define the sets Il ⊂ {1, , n}l k − ≥ l ≥ by Il−1 : Tl Il , k ≥ l ≥ 1.7 By 1.6 , I1 {1, , n} and this implies that αI1 ,i for ≤ i ≤ n From 1.6 , again, we have αIl ,i ≥ k − ≥ l ≥ 1, ≤ i ≤ n For every k ≥ l ≥ and for any j1 , , jl−1 ∈ Il−1 , let HIl j1 , , jl−1 : i1 , , il , m ∈ Il × {1, , l} | Sl,m i1 , , il j1 , , jl−1 1.8 Journal of Inequalities and Applications Using these sets we define the functions tIk ,l : Il → N k ≥ l ≥ inductively by tIk ,k i1 , , ik : 1, i , , ik ∈ I k , tIk ,l−1 j1 , , jl−1 : 1.9 tIk ,l i1 , , il i1 , ,il ,m ∈HIl j1 , ,jl−1 p1 , , pn such that Let J be an interval in R, let x : x1 , , xn ∈ J n , let p : n pi > ≤ i ≤ n and i pi 1, and let f : J → R be a convex function For any k ≥ l ≥ 1, set Al,l l Al,l Ik ; x; p : i1 , ,il ∈Il s f l s pis /αIl ,is xis l s pis /αIl ,is f pis α Il ,is l s pis /αIk ,is xis l s pis /αIk ,is , 1.10 and associate to each k − ≥ l ≥ the number Ak,l Ik ; x; p Ak,l : k−1 l tIk ,l i1 , , il i1 , ,il ∈Il s pis αIk ,is 1.11 We need the following hypotheses H2 Let x : x1 , , xn and p : p1 , , pn be positive n-tuples such that n i pi H3 Let J ⊂ R be an interval, let x : x1 , , xn ∈ J , let p : p1 , , pn be a positive 1, and let h, g : J → R be continuous and strictly n-tuples such that n pi i monotone functions n H4 Let J ⊂ R be an interval, let x : x1 , , xn ∈ J n , and let p : p1 , , p2 be positive n-tuples such that ni pi Further, let f : J → R be a convex function p Assume H1 and H2 The power means of order r ∈ R corresponding to il : i1 , , in ∈ I1 l 1, , k are given as Mr Ik , il : ⎧⎛ ⎪ ⎪ ⎪⎝ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ l s pis /αIk ,is xirs l s ⎞1/r ⎠ pis /αIk ,is , r / 0, 1.12 l s l s 1/ pi /αIk ,is pis /αIk ,is xis s , r We also use the means ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Mr : ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1/r n pi xir , r / 0, i n i p xi i , 1.13 r 4 Journal of Inequalities and Applications For γ, η ∈ R, we introduce the mixed symmetric means with positive weights as follows: ⎧⎡ ⎪ ⎪ ⎪ ⎪⎣ ⎪ ⎪ ⎪ ⎨ Mη,γ Ik , k : ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩k i k ik i1 , ,ik ∈Ik s ⎤ 1/η pis αIk ,is η Mγ I k , i k ⎦ , η / 0, 1.14 k s Mγ I k , i k pis /αIk ,is , η 0, i1 , ,ik ∈Ik and, for k − ≥ l ≥ 1, Mη,γ Ik , l : ⎧⎡ ⎪ ⎪ ⎪ ⎪⎣ ⎪ ⎪ k − l ⎪ ⎪ il ⎪ ⎨ ⎪⎡ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎣ ⎪ ⎪ ⎩ il l tIk ,l il i1 , ,il ∈Il s ⎤1/η pis α Ik ,is Mγ I k , i l η ⎦ , η / 0, ⎤1/ k−1 l Mγ I k , i l tIk ,l i l l s pis /αIk ,is ⎦ , η i1 , ,il ∈Il 1.15 We deduce the monotonicity of these means from the following refinement of the discrete Jensen inequality in Theorem 1.1 Assume (H1 ) and (H4 ) Then, n f pi xi ≤ Ak,k ≤ Ak,k−1 ≤ · · · ≤ Ak,2 ≤ Ak,1 i n pi f xi , 1.16 i where the numbers Ak,l k ≥ l ≥ are defined in 1.10 and 1.11 If f is a concave function, then the inequalities in 1.16 are reversed Under the conditions of the previous theorem, Υ1 x, p, f : Ak,m − Ak,l ≥ 0, Υ2 x, p, f : Ak,l − f n pi xi k ≥ l > m ≥ 1, ≥ 0, k ≥ l ≥ 1.17 i Corollary 1.2 Assume (H1 ) and (H2 ) Let η, γ ∈ R such that η ≤ γ, then Mγ 1 Mγ,η Ik , ≥ · · · ≥ Mγ,η Ik , k ≥ Mη , 1.18 Mη 1 Mη,γ Ik , ≤ · · · ≤ Mη,γ Ik , k ≤ Mγ 1.19 Journal of Inequalities and Applications Proof Assume η, γ / To obtain 1.18 , we can apply Theorem 1.1 to the function f x η η xγ/η x > and the n-tuples x1 , , xn to get the analogue of 1.16 and to raise the power 1/γ Equation 1.19 can be proved in a similar way by using f x xη/γ x > γ γ and x1 , , xn and raising the power 1/η When η or γ 0, we get the required results by taking limit Assume H1 and H3 Then, we define the quasiarithmetic means with respect to 1.10 and 1.11 as follows: ⎛ Mh,g k −1 ⎝ Ik , k : h i1 , ,ik ∈Ik s pis αIk ,is h◦g −1 k s ⎞ pis /αIk ,is g xis k s pis /αIk ,is ⎠, 1.20 and, for k − ≥ l ≥ 1, ⎛ Mh,g Ik , l h−1 ⎝ k − l il l tIk ,l il i1 , ,il ∈Il s pis α Ik ,is h ◦ g −1 l s ⎞ pis /αIk ,is g xis l s pis /αIk ,is ⎠ 1.21 The monotonicity of these generalized means is obtained in the next corollary Corollary 1.3 Assume (H1 ) and (H3 ) For a continuous and strictly monotone function q : J → R, one defines Mq : q−1 n pi q xi 1.22 i Then, Mh 1 Mh,g Ik , ≥ · · · ≥ Mh,g Ik , k ≥ Mg , 1.23 if either h ◦ g −1 is convex and h is increasing or h ◦ g −1 is concave and h is decreasing, Mg 1 Mg,h Ik , ≤ · · · ≤ Mg,h Ik , k ≤ Mh , 1.24 if either g ◦ h−1 is convex and g is decreasing or g ◦ h−1 is concave and g is increasing Proof First, we can apply Theorem 1.1 to the function h ◦ g −1 and the n-tuples g x1 , , g xn , then we can apply h−1 to the inequality coming from 1.16 This gives 1.23 A similar argument gives 1.24 : g ◦ h−1 , h x1 , , h xn and g −1 can be used Throughout Examples 1.4-1.5, 1.9–1.12, which are based on examples in , the conditions H2 , in the mixed symmetric means, and H3 , in the quasiarithmetic means, will be assumed 6 Journal of Inequalities and Applications Example 1.4 Suppose i1 , i2 ∈ {1, , n}2 | i1 |i2 , I2 : where i1 |i2 means that i1 divides i2 Since i|i i n i αI2 ,i 1.25 1, , n , therefore 1.6 holds We note that d i , i 1, , n, 1.26 where n/i is the largest positive integer not greater than n/i, and d i means the number of positive divisors of i Then, 1.14 gives for η, γ ∈ R Mη,γ ⎧⎡ ⎪ ⎪ ⎪ ⎪⎣ ⎪ ⎪ ⎨ I2 , 2 i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ i1 ,i2 ∈I2 s n/is Mγ I , i i2 ⎤1/n pi s s n Mγ I , i k d is ⎦ , η / 0, 1.27 pi s / n/is d is η / 0, i1 ,i2 ∈I2 while 1.20 gives ⎛ Mh,g I2 , 2 h−1 ⎝ i1 ,i2 ∈I2 s pis n/is h ◦ g −1 d is ⎞ s pis / n/is s d is pis / n/is g xis d is ⎠ 1.28 Assume H4 holds, and consider the set I2 in Example 1.4 Then, Theorem 1.1 implies that n pr xr f ≤ i1 ,i2 ∈I2 r pis s n/is f d is s pis / s pis / n/is d is n/is xis n ≤ d is pr f xr , r 1.29 and thus Υ3 x, p, f : i1 ,i2 ∈I2 Υ4 x, p, f : n r s pis n/is d is pr f xr − i1 ,i2 ∈I2 s s pis / s pis / f pis n/is d is f n/is n/is d is xis d is s pis / n/is s pis / n/is n −f pr xr ≥ 0, r d is xis d is ≥ 1.30 Journal of Inequalities and Applications c n Pk , ,cn Example 1.5 Let ci ≥ be an integer i 1, , n , let k : i ci , and also let Ik consist of all sequences i1 , , ik in which the number of occurrences of i ∈ {1, , n} is ci i 1, , n Obviously, 1.6 holds, and, by simple calculations, we have n Ik−1 i c1 , ,c Pk−1 i−1 ,ci −1,ci Moreover, tIk ,k−1 i1 , , ik−1 , ,cn , αIk ,i k! ci , c1 ! · · · cn ! i 1, , n 1.31 k for c1 , ,c i1 , , ik−1 ∈ Pk−1 i−1 ,ci −1,ci , ,cn , i 1.32 1, , n Under the above settings, 1.15 can be written as Mη,γ Ik , k − ⎧⎡ ⎤ η/γ 1/η γ γ n ⎪ ⎪ pr xr − pi /ci xi ⎪⎣ n r ⎪ ⎦ , ⎪ c i − pi ⎪ k−1 ⎪ − pi /ci ⎪ i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎛ ⎪ ⎞1/ k−1 ⎪ ⎨ ci −pi /γ γ γ n n pr xr − pi /ci xi r ⎠ , ⎪⎝ ⎪ − pi /ci ⎪ i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n 1/ k−1 ci ⎪ n ⎪ ⎪ ⎪ −pi pr ⎪ xi xr , ⎩ i η / 0, γ / 0, γ / 0, η 0, γ 0, 0, η i 1.33 while 1.21 becomes Mh,g Ik , k − −1 h n r 1 n ci − pi h ◦ g −1 k−1i pr g xr − pi /ci g xi − pi /ci 1.34 Assume H4 holds, and consider the set Ik in Example 1.5 Then, Theorem 1.1 yields that Ak,k−1 n c i − pi f k−1i n r pr xr − pi /ci xi − pi /ci , 1.35 n f pr xr ≤ Ak,k−1 ≤ r n pr f xr r This shows that Υ5 x, p, f : Ak,k−1 − f n pr xr ≥ 0, r Υ x, p, f : n r pr f xr − Ak,k−1 ≥ 1.36 Journal of Inequalities and Applications The following result is also given in Theorem 1.6 Assume (H1 ) and (H4 ), and suppose |HII j1 , , jl−1 | Il−1 k ≥ l ≥ Then, Ak,l n l|Il | Al,l l s pis xis l s pis l f pis i1 , ,il ∈Il s βl−1 for any j1 , , jl−1 ∈ k ≥ l ≥ 1, 1.37 pr f xr , 1.38 and thus n f pr xr n ≤ Ak,k ≤ Ak−1,k−1 ≤ · · · ≤ A2,2 ≤ A1,1 r r If f is a concave function then the inequalities 1.38 are reversed Under the conditions of the previous theorem, we have, from 1.38 , that Υ7 x, p, f : Am,m − Al,l ≥ 0, k ≥ l > m ≥ 1, n Υ8 x, p, f : Al,l − f ≥ 0, pr xr 1.39 k ≥ l ≥ r Assume H1 and H2 , and suppose |HII j1 , , jl−1 | βl−1 for any j1 , , jl−1 ∈ Il−1 k ≥ l ≥ In this case, the power means of order r ∈ R corresponding to il : i1 , , il ∈ Il l 1, , k has the form Mr Il , il ⎧⎛ ⎪ ⎪ ⎪⎝ ⎪ ⎪ ⎪ ⎪ ⎨ Mr Ik , il l s pis xirs l s ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎞1/r ⎠ , pis 1.40 1/ l r / 0, l s pis xis pis , r s Now, for γ, η ∈ R and k ≥ l ≥ 1, we introduce the mixed symmetric means with positive weights related to 1.37 as follows: Mη,γ Il : ⎧⎡ ⎪ ⎪ n ⎪⎣ ⎪ ⎪ ⎪ l|I | ⎪ ⎪ l il ⎨ ⎪⎡ ⎪ ⎪ ⎪ ⎪ ⎪⎣ ⎪ ⎪ ⎩ il ⎤1/η l pis i1 , ,il ∈Il Mγ I l , i l η ⎦ , η / 0, s 1.41 ⎤n/l|Il | Mγ I l , i l i1 , ,il ∈Il l s pis ⎦ , η Journal of Inequalities and Applications Corollary 1.7 Assume (H1 ) and (H2 ), and suppose |HII j1 , , jl−1 | Il−1 k ≥ l ≥ Let η, γ ∈ R such that η ≤ γ Then, βl−1 for any j1 , , jl−1 ∈ Mγ 2 Mγ,η I1 ≥ · · · ≥ Mγ,η Ik ≥ Mη , Mη 2 Mη,γ I1 ≤ · · · ≤ Mη,γ Ik ≤ Mγ 1.42 Proof The proof comes from Corollary 1.2 Assume H1 and H3 , and suppose |HII j1 , , jl−1 | βl−1 for any j1 , , jl−1 ∈ Il−1 k ≥ l ≥ We define for k ≥ l ≥ the quasiarithmetic means with respect to 1.37 as follows: ⎛ Mh,g n Il : h−1 ⎝ l|Il | l pis i1 , ,il ∈Il h ◦ g −1 l s l s s Corollary 1.8 Assume (H1 ) and (H3 ), and suppose |HII j1 , , jl−1 | Il−1 k ≥ l ≥ Then, Mh ⎞ pis g xis pis ⎠ 1.43 βl−1 for any j1 , , jl−1 ∈ 2 Mh,g I1 ≥ · · · ≥ Mh,g Ik ≥ Mg , 1.44 where either h ◦ g −1 is convex and h is increasing or h ◦ g −1 is concave and h is decreasing, Mg 2 Mg,h I1 ≤ · · · ≤ Mg,h Ik ≤ Mh , 1.45 where either g ◦ h−1 is convex and g is decreasing or g ◦ h−1 is concave and g is increasing Proof The proof is a consequence of Corollary 1.3 Example 1.9 If we set Ik : i1 , , ik ∈ {1, , n}k | i1 < · · · < ik , ≤ k ≤ n, 1.46 then αIn ,i i 1, , n , that is, 1.6 is satisfied for k n It comes easily that Tk Ik n Ik−1 k 2, , n , |Ik | k 1, , n , and for each k 2, , n k HIk j1 , , jk−1 n− k−1 , j1 , , jk−1 ∈ Ik−1 1.47 10 Journal of Inequalities and Applications In this case, 1.41 becomes for n ≥ k ≥ ⎧⎡ ⎪ ⎪ ⎪⎢ ⎪ ⎪⎣ ⎪ ⎪ ⎪ ⎪ ⎨ Mη,γ Ik ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎤1/η k n−1 k−1 1≤i1