Abstract
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 Cebysev type inequalities. Mean value theorems and n-exponential convexity are discussed for functionals related to the new majorization identities.
Highlights
The aim of this paper is to present a new generalization of weighted majorization theorem for n-convex functions, by using generalization of Taylor’s formula
Author details 1Department of applied mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, Zagreb, 10000, Croatia. 2Department of Mathematical Sciences, Faculty of Science, University of Karachi, University Road, Karachi, 75270, Pakistan. 3Faculty of textile technology, University of Zagreb, Prilaz baruna Filipovica 28A, Zagreb, 10000, Croatia
Summary
For every continuous convex function f : [a, b] → R, the following inequality holds: wif (xi) ≤ wif (yi). Remark Under the assumptions of Proposition , for every concave function f the reverse inequality holds in For every n-convex function f : I → R the following inequality holds: wif (yi) – wif (xi). Its integral analogs are given as follows: Theorem Let all the assumptions of Theorem hold with the additional condition β β w(t)Tn x(t), s dt ≤ w(t)Tn y(t), s dt, ∀s ∈ [a, b], where Tn(·, s) is defined in Proposition. We are ready to state the main results of this section: Theorem Let n ∈ N, f : [a, b] → R be such that f (n) is an absolutely continuous function with (· – a)(b – ·)[f (n+ )] ∈ L[a, b] and xi, yi ∈ [a, b], wi ∈ R
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