Generalizations of Sherman’s inequality by Lidstone’s interpolating polynomial

Ravi P Agarwal,Slavica Ivelić Bradanović,Josip Pečarić

doi:10.1186/s13660-015-0935-6

Ravi P Agarwal, Slavica Ivelić Bradanović + Show 1 more

Open Access

https://doi.org/10.1186/s13660-015-0935-6

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Abstract

In majorization theory, the well-known majorization theorem plays a very important role. A more general result was obtained by Sherman. In this paper, concerning 2n-convex functions, we get generalizations of these results applying Lidstone’s interpolating polynomials and the Cebysev functional. Using the obtained results, we generate a new family of exponentially convex functions. The results are some new classes of two-parameter Cauchy type means.

Highlights

For fixed m ≥, let x = (x, . . . , xm) and y = (y, . . . , ym) denote two m-tuples
The following notion of Schur-convexity generalizes the definition of a convex function via the notion of majorization
4 Mean value theorems and exponential convexity we present mean-value theorems of Lagrange and Cauchy type using results from the previous section

Summary

Introduction

The following notion of Schur-convexity generalizes the definition of a convex function via the notion of majorization. ). Theorem (Majorization theorem) Let I ⊂ R be an interval and x = Ym) ∈ Im. Let f : I → R be continuous function. The following theorem gives a weighted generalization of the majorization theorem ). Theorem (Fuchs’ theorem) Let x = Ym) ∈ Im be two decreasing m-tuples and p = Pm) be a real m-tuple such that k k piyi ≤ pixi, i=. For every continuous convex function f : I → R, we have m m pif (yi) ≤ pif (xi)

The Jensen inequality in the form f m

Proof Let us denote

Then for ε small enough β

Since d n ψt dx n etx

Conclusions