Search

This paper gives extensions and improvements of Sherman’s inequality forn-convex functions obtained by using new identities which involve Green’s functions and Fink’s identity. Moreover, extensions and improvements of Majorization inequality as well as Jensen’s inequality are obtained as direct consequences. New inequalities between geometric, logarithmic and arithmetic means are also established.

New generalizations of Sherman's inequality for n-convex functions are obtained using Fink's identity and Green's function. Using inequalities for the Čebyšev functional, some new Ostrowsky and Grüss type inequalities related to these generalizations are established.

The matrix convexity and the matrix monotony of a real $C^1$ function $f$ on $(0,\infty)$ are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with $f$, $tf(t)$, and $t^2f(t)$. Similar characterizations are also obtained for matrix monotone functions on a finite interval $(a,b)$., 20 pages