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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 163202, 10 pages doi:10.1155/2008/163202 Research Article New Means of Cauchy’s Type Matloob Anwar 1 and J. Pe ˇ cari ´ c 1, 2 1 Abdus Salam School of Mathematical Sciences, GC University, Lahore Gulberg 54660, Pakistan 2 Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia Correspondence should be addressed to Matloob Anwar, matloob t@yahoo.com Received 30 December 2007; Accepted 7 April 2008 Recommended by Wing-Sum Cheung We will introduce new means of Cauchy’s type M s r,l f, μ defined, for example, as M s r,l f, μ ll −s/rr − sM r r f, μ − M r s f, μ/M l l f, μ − M l s f, μ 1/r−l , in the case when l / r / s, l, r / 0. We will show that this new Cauchy’s mean is monotonic, that is, the following result. Theorem.Let t, r, u, v ∈ R,suchthatt ≤ v, r ≤ u.ThenforM s r,l f, μ, one has M s t,r ≤ M s v,u . We will also give some related comparison results. Copyright q 2008 M. Anwar and J. 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. 1. Introduction Let Ω be a convex set equipped with a probability measure μ. Then for a strictly monotonic continuous function f, the integral power mean of order r ∈ R is defined as follows: M r f, μ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Ω fu r dμu 1/r ,r / 0, exp Ω log fu dμu ,r 0. 1.1 Throughout our present investigation, we tacitly assume, without further comment, that all the integrals involved in our results exist. The following inequality for differences of power means was obtained see 1, Remark 8: rr − s ll − s m ≤ M r r f, μ − M r s f, μ M l l f, μ − M l s f, μ ≤ rr − s ll − s M, 1.2 2 Journal of Inequalities and Applications where r, l, s ∈ R, l / r / s, r,l / 0andwherem and M are, respectively, the minimum and the maximum values of the function x r−l on the image of fuu ∈ Ω. Letusnotethat1.2 was obtained as consequence of the following result see, e.g., 1, Corollary 1. Theorem 1.1. Let r, s, l ∈ R, and let Ω be a convex set equipped with a probability measure μ.Then, M r r f, μ − M r s f, μ M l l f, μ − M l s f, μ rr − s ll − s η r−l 1.3 for some η in the image of fuu ∈ Ω, provided that the denominator on the left-hand side of 1.3 is non-zero. We can also note that from 1.3 we can get the following form of 1.2: inf u∈Ω fu ≤ ll − s rr − s M r r f, μ − M r s f, μ M l l f, μ − M l s f, μ 1/r−l ≤ sup u∈Ω fu, 1.4 where r, l, s ∈ R, r / l / s, r, l / 0. Moreover, 1.4 suggests introducing a new mean of Cauchy type. We will prove in Section 3 a comparison theorem for these means. Finally we will, in Section 4, give some applications. 2. New Cauchy’s mean From 1.4, we can define a new mean M s r,l as follows: M s r,l f, μ ll − s rr − s M r r f, μ − M r s f, μ M l l f, μ − M l s f, μ 1/r−l ,l / r / s, l, r / 0. 2.1 Now by taking lim l→0 M s r,l f, μ, we will get M s r,0 f, μM s 0,r f, μlim l→0 M s r,l f, μ s M r r f, μ − M r s f, μ rr − s log M s f, μ − log M 0 f, μ 1/r ,r / s, r, s / 0. 2.2 Now by taking lim r→s M s r,l f, μ, we will get lim r→s M s r,l f, μM s s,l f, μM s l,s f, μ ll − s s fu s log fudμu−M s s f, μ log M s f, μ M l l f, μ−M l s f, μ 1/s−l ,l / s, l, s / 0. 2.3 M. Anwar and J. Pe ˇ cari ´ c3 By similar way, we can calculate all the cases for r, s, l ∈ R. Finally, we get the following definition of M s r,l f, μ: M s r,l f, μ ll − s rr − s M r r f, μ − M r s f, μ M l l f, μ − M l s f, μ 1/r−l ,l / r / s, l, r / 0; M s r,0 f, μM s 0,r f, μ s M r r f, μ − M r s f, μ rr − s log M s f, μ − log M 0 f, μ 1/r ,r / s, r, s / 0; M s s,l f, μM s l,s f, μ ll − s s fu s log fudμu − M s s f, μ log M s f, μ M l l f, μ − M l s f, μ 1/s−l , l / s, l, s / 0; M s s,0 f, μM s 0,s f, μ fu s log fudμu − M s s f, μ log M s f, μ log M s f, μ − log M 0 f, μ 1/s ,s / 0; M 0 r,l f, μ l 2 M r r f, μ − M r 0 f, μ r 2 M l l f, μ − M l 0 f, μ 1/r−l ,l,r / 0; M 0 r,0 f, μM 0 0,r f, μ 2 M r r f, μ − M r 0 f, μ r 2 M 2 2 log f, μ − M 2 1 log f,μ 1/r ,r / 0; M s t,t exp − 2t − s tt − s f t log fdμu − M t s f, μ log M s f, μ M t t f, μ − M t s f, μ ,t / s; M 0 t,t exp − 2 t f t log fdμu − M t 0 f, μ log M 0 f, μ M t t f, μ − M t 0 f, μ ,t / 0; M 0 0,0 exp 1 3 log f 3 dμu − log M 0 f, μ 3 log f 2 dμu − log M 0 f, μ 2 , M s s,s exp − 1 s f s log f 2 dμu − M s s f, μ log M s f, μ 2 2 f s log fdμu − M s s f, μ log M s f, μ ,s / 0; M s 0,0 exp 1 s log f 2 dμu − log M s f, μ 2 2 log fdμu − log M s f, μ ,s / 0. 2.4 3. Monotonicity of new means In this section, we will prove the monotonicity of 2.4. We need the following lemmas for log-convex function. 4 Journal of Inequalities and Applications Lemma 3.1. Let f be log-convex function and if x 1 ≤ y 1 ,x 2 ≤ y 2 ,x 1 / x 2 ,y 1 / y 2 , then the following inequality is valid: f x 2 f x 1 1/x 2 −x 1 ≤ f y 2 f y 1 1/y 2 −y 1 . 3.1 Proof. In 2, page 3 we have the following result for convex function f,withx 1 ≤ y 1 ,x 2 ≤ y 2 ,x 1 / x 2 ,y 1 / y 2 : f x 2 − f x 1 x 2 − x 1 ≤ f y 2 − f y 1 y 2 − y 1 . 3.2 Putting f log f, we get log f x 2 f x 1 1/x 2 −x 1 ≤ log f y 2 f y 1 1/y 2 −y 1 , 3.3 after applying exponential function we get 3.1. The following two lemmas are proved for functionals in 3Theorem 4 and Lemma 2, for Lemma 3.2 see also 4,Theorem1. Lemma 3.2. Let us consider Λ t defined as Λ t g,μ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ M t t g,μ − M t 1 g,μ tt − 1 ,t / 0, 1; log M 1 g,μ − log M t 0 g,μ,t 0; g log gμ − M 0 g,μ log M 0 g,μ,t 1. 3.4 Then, Λ t is a log-convex function. Lemma 3.3. Let us consider Λ t defined as Λ t ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 t 2 M t t f, μ − M t 0 f, μ ,t / 0; 1 2 M 2 2 log f, μ − M 2 1 log f, μ ,t 0. 3.5 Then, Λ t is a log-convex function. Theorem 3.4. Let t, r, u, v ∈ R,suchthat,t ≤ v, r ≤ u. Then for 2.4,wehave M s t,r ≤ M s v,u . 3.6 M. Anwar and J. Pe ˇ cari ´ c5 Proof Case 1 s / 0. Let us consider Λ t defined as in Lemma 3.2. Λ t is a continuous and log-convex. So, Lemma 3.1 implies that for t, r, u, v ∈ R, such that, t ≤ v, r ≤ u, t / r, v / u,wehave Λ t Λ r 1t−r ≤ Λ v Λ u 1/v−u . 3.7 For s>0 by substituting g f s ,t t/s, r r/s, u u/s, v v/s ∈ R, such that, t/s ≤ v/s, r/s ≤ u/s, t / r, v / u,in3.4,weget Λ t,s f, μ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s 2 t1 − s M t t f, μ − M t s f, μ ,t / 0,s; s log M s f, μ − log M 0 f, μ ,t 0; s f s log f − M s 0 f, μ log M 0 f, μ ,t s. 3.8 And 3.7 becomes Λ t,s Λ r,s 1t−r ≤ Λ v,s Λ u,s 1/v−u . 3.9 From 3.9, we get our required result. Now when s<0 by substituting g f s ,t t/s, r r/s, u u/s, v v/s ∈ R, such that, v/s ≤ t/s, u/s ≤ r/s, t / r, v / u,in3.4 we get 3.8. And 3.7 becomes Λ v,s Λ u,s s/v−u ≤ Λ t,s Λ r,s s/t−r . 3.10 Now s<0, from 3.10, by raising power −s,weget Λ t,s Λ r,s 1/t−r ≤ Λ v,s Λ u,s 1/v−u . 3.11 From 3.11, we get our required result. Case 2 s 0. In this case, we can get our result by taking limit s→0in3.8 and also in this case we can consider Λ t defined as in Lemma 3.3. Λ t is log-convex function. So, Lemma 3.1 implies that for t, r, u, v ∈ R, such that, t ≤ v, r ≤ u, t / r, v / u,wehave Λ t Λ r 1/t−r ≤ Λ v Λ u 1/v−u . 3.12 Therefore, we have for t, r, u, v ∈ R, such that, t ≤ v, r ≤ u, t / r, v / u: M 0 t,r ≤ M 0 v,u , 3.13 which completes the proof. 6 Journal of Inequalities and Applications 4. Further consequences and applications In this section, we will represent the various applications of our previous definition of a new Cauchy mean and monotonicity of this above defined a new Cauchy mean. 4.1. Tobey and Stolarsky-Tobey means Let E n−1 represent the n − 1-dimensional Euclidean simplex given by E n−1 u 1 ,u 2 , ,u n−1 : u i ≥ 0, 1 ≤ i ≤ n − 1, n−1 i1 u i ≤ 1 , 4.1 and set u n 1 − n−1 i1 u i . Moreover, with u u 1 , ,u n , let μu be a probability measure on E n−1 . The power mean of order p p ∈ R of the positive n-tuple x x 1 , ,x n ∈ R n ,withthe weights u u 1 , ,u n , is defined by M p x, μ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n i1 u i x p i 1/p ,p / 0; n i1 x u i i ,p 0. 4.2 Then, the Tobey mean L p,r x; μ is defined as follows: L p,r x; μM r M p x, μ; μ , 4.3 where M r g,μ denotes the integral power mean, in which Ω is now the n − 1-dimensional Euclidean simplex E n−1 . We note that, since M p x, μ isameanwehavemin{x i }≤M p x, μ ≤ max{x i }. Now setting fx, μM p x, μ in 2.4 we get Γ s p,r,l x, μ ll − s rr − s L r p,r x, μ − L r p,s x, μ L l p,l x, μ − L l p,s x, μ 1/r−l ,l / r / s, l, r / 0; Γ s p,r,0 x, μΓ s p,0,r x, μ s L r p,r x, μ − L r p,s x, μ rr − s log L p,s x, μ − log L p,0 x, μ 1/r ,r / s, r, s / 0; Γ s p,s,l x, μΓ s p,l,s x, μ ll − s s M p x, μ s log dμu − L s p,s x, μ log L p,s x, μ L l p,l x, μ − L l p,s x, μ 1/s−l , l / s, l, s / 0; Γ s p,s,0 x, μΓ s p,0,s x, μ M p x, μ s log M p x, μdμu−L s p,s x, μ log L p,s x, μ log L p,s x, μ−log L p,0 x, μ 1/s ,s / 0; M. Anwar and J. Pe ˇ cari ´ c7 Γ 0 p,r,l x, μ l 2 L r p,r x, μ − L r p,0 x, μ r 2 L l p,l x, μ − L l p,0 x, μ 1/r−l ,l,r / 0; Γ 0 p,r,0 x, μΓ 0 p,0,r x, μ 2 L r p,r x, μ − L r p,0 x, μ r 2 M 2 2 log M p x, μ,μ − M 2 1 log M p x, μ,μ 1/r ,r / 0; Γ s p,t,t x, μexp − 2t − s tt − s M p x, μ t log M p x, μdμu−L t p,s x, μ log L p,s x, μ L t p,t x, μ−L t p,s x, μ ,t / s; Γ 0 p,t,t x, μexp − 2 t M p x, μ t log M p x, μdμu − L t p,0 x, μ log L p,0 x, μ L t p,t x, μ − L t p,0 x, μ ,t / 0; Γ 0 p,0,0 x, μexp 1 3 log M p x, μ 3 dμu − log L p,0 x, μ 3 log M p x, μ 2 dμu − log L p,0 x, μ 2 , Γ s p,s,s x, μexp − 1 s M p x, μ s log M p x, μ 2 dμu−L s p,s x, μ log L p,s x, μ 2 2 M p x, μ s log M p x, μdμu− L s p,s x, μ log L p,s x, μ ,s / 0; Γ s p,0,0 x, μexp 1 s log M p x, μ 2 dμu − log L p,s x, μ 2 2 log M p x, μdμu − log M s x, μ ,s / 0. 4.4 Theorem 4.1. Let t, r, u, v ∈ R,suchthat,t<v, r<u.Then for 4.4,wehave Γ s p,t,r ≤ Γ s p,v,u . 4.5 Proof. It is a simple consequence of Theorem 3.4. Pe ˇ cari ´ cand ˇ Simi ´ c see 5, Definition 1 introduced the Stolarsky-Tobey mean ε p,q x, μ defined by ε p,q x, μL p,q−p x, νM q−p M p x, μ; μ , 4.6 where L p,r x, ν is the Tobey mean already introduced above. For the Stolarsky-Tobey mean and 2.4,wegetthefollowing: Υ s p,r,l x, μ ll − s rr − s ε r p,pr x, μ − ε r p,ps x, μ ε l p,pl x, μ − ε l p,ps x, μ 1/r−l ,l / r / s, l, r / 0; Υ s p,r,0 x, μΥ s p,0,r x, μ s ε r p,pr x, μ − ε r p,ps x, μ rr − s log ε p,ps x, μ − log ε p,p x, μ 1/r ,r / s, r, s / 0; 8 Journal of Inequalities and Applications Υ s p,s,l x, μΥ s p,l,s x, μ ll − s s M p x, μ s log dμu − ε s p,ps x, μ log ε p,ps x, μ ε l p,pl x, μ − ε l p,ps x, μ 1/s−l , l / s, l, s / 0; Υ s p,s,0 x, μΥ s p,0,s x, μ M p x, μ s log M p x, μdμu − ε s p,ps x, μ log ε p,ps x, μ log ε p,ps x, μ − log ε p,p x, μ 1/s , s / 0; Υ 0 p,r,l x, μ l 2 ε r p,pr x, μ − ε r p,p x, μ r 2 ε l p,pl x, μ − ε l p,p x, μ 1/r−l ,l,r / 0; Υ 0 p,r,0 x, μΥ 0 p,0,r x, μ 2ε r p,pr x, μ − ε r p,p x, μ r 2 M 2 2 log M p x, μ,μ − M 2 1 log M p x, μ,μ 1/r ,r / 0; Υ s p,t,t x, μexp − 2t − s tt − s M p x, μ t log M p x, μdμu − M t s log ε p.ps x, μ ε t p,pt x, μ − ε t p,ps x, μ ,t / s; Υ 0 p,t,t x, μexp − 2 t M p x, μ t log M p x, μdμu − ε t P,p x, μ log ε p,p x, μ ε t p,pt x, μ − ε t p,p x, μ ,t / 0; Υ 0 p,0,0 x, μexp 1 3 log M p x, μ 3 dμu − log ε p,p x, μ 3 log M p x, μ 2 dμu − log ε p,p x, μ 2 , Υ s p,s,s x, μexp − 1 s M p x, μ s log M p x, μ 2 dμu − ε s p,ps x, μlog ε p.ps x, μ 2 2 M p x, μ s log M p x, μdμu − ε s p.ps x, μ log ε p,ps x, μ , s / 0; Υ s p,0,0 x, μexp 1 s log M p x, μ 2 dμu − log ε p,ps x, μ 2 2 log M p x, μdμu − log ε p.ps x, μ ,s / 0. 4.7 Theorem 4.2. Let t, r, u, v ∈ R,suchthat,t<v, r<u.Then for 4.7,wehave Υ s p,t,r ≤ Υ s p,v,u . 4.8 Proof. It is a simple consequence of Theorem 3.4. M. Anwar and J. Pe ˇ cari ´ c9 4.2. The complete symmetric mean The rth complete symmetric polynomial mean the complete symmetric mean of the positive real n-tuple x is defined by see 6, pages 332,341 Q r n x q r n x 1/r c r n x nr−1 r 1/r , 4.9 where c 0 n x1andc r n x n j1 n i1 x i j i and the sum is taken over all nr−1 r nonnegative integer n-tuples i 1 , ,i n with n j1 i j r r / 0. The complete symmetric polynomial mean can also be written in an integral form as follows: Q r n E n−1 n i1 x i u i r dμu 1/r , 4.10 where μ represents a probability measure such that dμun−1!du 1 ···du n−1 . We can see this as a special case of the integral power mean M r f, μ,wherefu n i1 x i u i ,μis a probability measure as above, and Ω is the above defined n−1-dimensional simplex E n−1 . Thus from 2.4, we have the following result: Θ s n,r,l x, μ ll − s rr − s Q r n r x, μ − Q s n r x, μ Q l n l x, μ − Q s n l x, μ 1/r−l ,l / r / s, l, r ∈ N. 4.11 A simple consequence of Theorem 3.4 is the following result. Theorem 4.3. Let t, r, u, v ∈ N,suchthat,t<v, r<u.Then for 4.11,wehave Θ s n,t,r ≤ Θ s n,v,u . 4.12 4.3. Whiteley means Let x be a positive real n-tuple, s ∈ R s / 0 and r ∈ N. Then, the sth function of degree r is defined by the following generating function see 6, pages 341–344: ∞ r0 t r,s n xt r ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n i1 1 x i t s ,s>0, n i1 1 − x i t s ,s<0. 4.13 The Whiteley mean is now defined by W r,s n x w r,s n x 1/r ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t r,s n x nr s 1/r ,s>0, t r,s n x −1 r nr s 1/r ,s<0. 4.14 10 Journal of Inequalities and Applications For s<0, the Whiteley mean can be further generalized if we slightly change the definition of t r,s n x and define h r,σ n x as follows: ∞ r0 h r,σ n xt r n i1 1 1 − x i t σ i , 4.15 where σ σ 1 , ,σ n ; σ ∈ R ; i 1, ,n. The following generalization of the Whiteley mean for s<0 is defined by see 7, Lemma 2.3 H r,σ n x h r,σ n x n i1 σ i r−1 r 1/r . 4.16 If we denote by μ a measure on the simplex Δ n−1 {u 1 , ,u n : u i ≥ 0,i 1, ,n − 1, n i1 u i ≤ 1} such that dμu Γ n i1 σ i n i1 Γ σ i n i1 u σ i −1 i du 1 ···du n−1 , 4.17 where u n 1 − n−1 i−1 , then we have μ as a probability measure and we can also write the mean H r,σ n x in integral form as follows: H r,σ n x Δ n−1 n i1 x i u i r dμu 1/r . 4.18 Finally, just as we did above in this investigation, we can develop the following analogous definition: H s n,r,l x, μ ⎛ ⎜ ⎝ ll − s rr − s H r,σ n r x, μ − H s,σ n r x, μ H l,σ n l x, μ − H s,σ n l x, μ ⎞ ⎟ ⎠ 1/r−l ,l / r / s, l, r ∈ N. 4.19 A simple consequence of Theorem 3.4 is the following result. Theorem 4.4. Let t, r, u, v ∈ N,suchthat,t<v, r<u.Then for 4.19,wehave H s n,t,r ≤ H s n,v,u . 4.20 References 1 J. Pe ˇ cari ´ c, M. R. Lipanovi ´ c, and H. M. Srivastava, “Some mean-value theorems of the Cauchy type,” Fractional Calculus & Applied Analysis, vol. 9, no. 2, pp. 143–158, 2006. 2 J. Pe ˇ cari ´ c, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, vol. 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992. 3 M. Anwar and J. Pe ˇ cari ´ c, “On logarithmic convexity for differences of power means,” to appear in Mathematical Inequalities & Applications. 4 S. Simi ´ c, “On logarithmic convexity for differences of power means,” Journal of Inequalities and Applications, vol. 2007, Article ID 37359, 8 pages, 2007. 5 J. Pe ˇ cari ´ candV. ˇ Simi ´ c, “Stolarsky-Tobey mean in n variables,” Mathematical Inequalities & Applications, vol. 2, no. 3, pp. 325–341, 1999. 6 P. S. Bullen, Handbook of Means and Their Inequalities, vol. 560 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. 7 J. Pe ˇ cari ´ c, I. Peri ´ c, and M. R. Lipanovi ´ c, “Generalized Whiteley means and related inequalities,” to appear in Mathematical Inequalities & Applications. . Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 163202, 10 pages doi:10.1155/2008/163202 Research Article New Means of Cauchy’s Type Matloob Anwar 1 and. suggests introducing a new mean of Cauchy type. We will prove in Section 3 a comparison theorem for these means. Finally we will, in Section 4, give some applications. 2. New Cauchy’s mean From 1.4,. μ ,s / 0. 2.4 3. Monotonicity of new means In this section, we will prove the monotonicity of 2.4. We need the following lemmas for log-convex function. 4 Journal of Inequalities and Applications Lemma
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