8th Croatian Mathematical Congress

Generalization of Mercer's Inequality for Averages of convex Functions and Divided Differences

3 Jul 2024, 09:20

30m

D1 (Faculty of Economics and Business, J. J. Strossmayer University of Osijek)

Poster ANL: Analysis and its Applications Poster session

Speaker

Gorana Aras-Gazic (University of Zagreb Faculty of Architecture)

Description

By utilizing integral arithmetic mean $F$ defined as
$$F(x,y)=\{\begin{array}{cc}\frac{1}{y-x}{\int }_x^{y}f(t)dt,& x,y\in I,x\ne y,\\ f(x),& x=y\in I\end{array}$$ and applying a generalized form of Niezgoda's inequalty, we present new proofs concerning the $(m,n)$-convexity of the integral arithmetic mean function. This paper aims to contribute novel proofs to D. E. Wulbert's findings, which demonstrate that the integral arithmetic mean exhibits convexity on $I^2$ provided $f$ is convex on $I$.
Additionally, we establish an inequality for divided differences by employing the extended form of Niezgoda's inequality. Moreover, we demonstrate the convexity of the function defined by these divided differences.