Weighted majorization theorems via generalization of Taylor’s formula

Journal of Inequalities and Applications, Jun 2015

A new generalization of the weighted majorization theorem for n-convex functions is given, by using a generalization of Taylor’s formula. Bounds for the remainders in new majorization identities are given by using the Čebyšev type inequalities. Mean value theorems and n-exponential convexity are discussed for functionals related to the new majorization identities. MSC: 26D15, 26D20.

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Weighted majorization theorems via generalization of Taylor’s formula

Aljinovic´ et al. Journal of Inequalities and Applications Weighted majorization theorems via generalization of Taylor's formula Andrea Aglic´ Aljinovic´ 0 1 3 6 Asif R Khan 0 1 2 3 5 Josip E Pecˇaric´ 0 1 3 4 0 Road, Karachi , 75270 , Pakistan 1 University of Karachi, University 2 Department of Mathematical 3 Sciences, Faculty of Science 4 Faculty of textile technology, University of Zagreb, Prilaz baruna Filipovic ́a 28A, Zagreb , 10000 , Croatia 5 Department of Mathematical Sciences, Faculty of Science, University of Karachi , University Road, Karachi, 75270 , Pakistan 6 Department of applied mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb , Unska 3, Zagreb, 10000 , Croatia A new generalization of the weighted majorization theorem for n-convex functions is given, by using a generalization of Taylor's formula. Bounds for the remainders in new majorization identities are given by using the Cˇ ebyšev type inequalities. Mean value theorems and n-exponential convexity are discussed for functionals related to the new majorization identities. majorization theorem; Taylor's formula; Cˇ ebyšev functional 1 Introduction f λx + ( – λ)x ≤ λf (x) + ( – λ)f (x) (b) If the inequality in (.) is reversed, then f is called concave. If it is strict for each Proposition  A function f : I → R is convex if the inequality © 2015 Aljinovic´ et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The following result can be deduced from Proposition . Proposition  If a function f : I → R is convex, then the inequality Definition  The nth order divided difference of a function f : I → R at distinct points xi, xi+, . . . , xi+n ∈ I = [a, b] ⊂ R for some i ∈ N is defined recursively by [xj; f ] = f (xj), j ∈ {i, . . . , i + n}, [xi, . . . , xi+n; f ] = [xi+, . . . , xi+n; f ] – [xi, . . . , xi+n–; f ] . xi+n – xi It may easily be verified that [xi, . . . , xi+n; f ] = Remark  Let us denote [xi, . . . , xi+n; f ] by (n)f (xi). The value [xi, . . . , xi+n; f ] is independent of the order of the points xi, xi+, . . . , xi+n. We can extend this definition by including the cases in which two or more points coincide by taking respective limits. Definition  A function f : I → R is called convex of order n or n-convex if for all choices of (n + ) distinct points xi, . . . , xi+n we have (n)f (xi) ≥ . Remark  For n =  and i = , we get the second order divided difference of a function f : I → R, which is defined recursively by for arbitrary points x, x, x ∈ I. Now, we discuss some limiting cases as follows: taking the limit as x → x in (.), we get , x = x, [xj; f ] = f (xj), j ∈ {, , }, [xj, xj+; f ] = [x, x, x; f ] = , j ∈ {, }, provided that f (x) exists. For fixed m ≥ , let x = (x, . . . , xm) and y = (y, . . . , ym) denote two real m-tuples and x[] ≥ x[] ≥ · · · ≥ x[m], y[] ≥ y[] ≥ · · · ≥ y[m] their ordered components. Definition  For x, y ∈ Rm, This notion and notation of majorization was introduced by Hardy et al. []. Now, we state the well-known majorization theorem from the same book [] as follows. Proposition  Let x, y ∈ [a, b]m. The inequality f (xi) ≤ wixi ≤ wixi = wif (xi) ≤ wif (yi). The following weighted version of the majorization theorem was given by Fuchs in [] (see also [], p. and [], p.). wiyi, k ∈ {, . . . , m – } and Then for every continuous convex function f : [a, b] → R, the following inequality holds: Remark  Under the assumptions of Proposition , for every concave function f the reverse inequality holds in (.). w(t)x(t) dt ≤ w(t)y(t) dt, for each u ∈ (α, β), and w(t)x(t) dt = w(t)y(t) dt, w(t)f x(t) dt ≤ w(t)f y(t) dt. hold, then for every continuous convex function f : I → R the following inequality holds: w(t)x(t) dt ≤ w(t)y(t) dt, for each u ∈ (α, β), and w(t)x(t) dt = w(t)y(t) dt, The following proposition is a consequence of Theorem  in [] (see also [], p.) and represents an integral majorization result. α β α β then again inequality (.) holds. In this paper we will state our results for decreasing x and y satisfying the assumption of Proposition , but they are still valid for increasing x and t satisfying the above condition; see for example [], p.. Proposition  Let n ∈ N, f : I → R be such that f (n–) is absolutely continuous, I ⊂ R an open interval, a, b ∈ I, a < b. Then the following identity holds: f (x) = f (t) dt + k= k!(k + ) k= k!(k + ) Tn(x, s)f (n)(s) ds, ⎧ Tn(x, s) = ⎨ – n(x(b––s)an) + bx––aa (x – s)n–, a ≤ s ≤ x, ⎩ – n(x(b––s)an) + bx––ba (x – s)n–, x < s ≤ b. In case n =  the sum kn=– · · · is empty, so i (...truncated)


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Andrea Aljinović, Asif R Khan, Josip E Pečarić. Weighted majorization theorems via generalization of Taylor’s formula, Journal of Inequalities and Applications, 2015, pp. 196, 2015, DOI: 10.1186/s13660-015-0710-8