Speaker
Gorana Aras-Gazic
(University of Zagreb Faculty of Architecture)
Description
By utilizing integral arithmetic mean $F$ defined as
$$
F(x,y)=\left\{\begin{array}{cc}
\frac{1}{y-x}\int_{x}^{y}f(t)dt, & x,y\in I,~x\neq y,\\
f(x), & x=y\in I
\end{array}\right.
$$
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.
Author
Gorana Aras-Gazic
(University of Zagreb Faculty of Architecture)
Co-authors
Marjan Praljak
(University of Zagreb Faculty of Food Technology and Biotechnology)
Josip Pečarić
(Croatian Academy of Sciences and Arts)