Abstract
In this paper we give generalized results of a majorization inequality by using extension of the Montgomery identity and newly defined Green’s functions (Mehmood et al. in J. Inequal. Appl. 2017(1):108, 2017). We obtain a generalized majorization theorem for the class of n-convex functions. We use Csiszár f-divergence and generalized majorization-type inequalities to obtain new generalized results. We further discuss our obtained generalized results in terms of the Shannon entropy and the Kullback–Leibler distance.
Highlights
The theory of majorization is perhaps most remarkable for its simplicity
Easy-to-use, and flexible mathematical tool which can be applicable to a wide number of fields
Many important contributions were made by other authors
Summary
The theory of majorization is perhaps most remarkable for its simplicity. It is a powerful, easy-to-use, and flexible mathematical tool which can be applicable to a wide number of fields. X majorizes y if and only if the following inequality holds: m m f (yi) ≤ f (xi), (1) We present our results for nonincreasing functions φ and ψ which satisfy the conditions of Theorem 3, but those results hold too for nondecreasing φ and ψ satisfying the following inequality: b b p(w)ψ(w) dw ≤ p(w)φ(w) dw, for every λ ∈ [a, b], (8) Motivated by Abel–Gontscharoff Green’s function for ‘two-point right focal problem’, Mehmood et al (see [34]) presented some new types of Green’s functions which are continuous as well as convex, as follows: Let [ζ1, ζ2] ⊂ R.
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