Vector majorization and Schur-concavity of some sums generated by the Jensen and Jensen-Mercer functionals

Bounds for the Normalized Jensen – Mercer Functional

We introduce the normalized Jensen-Mercer functional Mn( f ,x, p) = f (a)+ f (b)− n ∑ i=1 pi f (xi)− f ( a+b− n ∑ i=1 pixi ) and establish the inequalities of type MMn( f ,x,q) Mn( f ,x, p) mMn( f ,x,q) , where f is a convex function, x = (x1, . . . ,xn) and m and M are real numbers satisfying certain conditions. We prove them for the case when p and q are nonnegative n -tuples and when p and q...

On a Variant of the Jensen–mercer Inequality for Operators

Some refinements of the Jensen-Mercer inequality for operators are presented. Obtained results are used to refine some comparision inequalities between power and quasiarithmetic means for operators. Mathematics subject classification (2000): 47A63, 47A64.

Some new estimates of the ‘Jensen gap’

holds. This inequality can be traced back to Jensen’s original papers [, ] and is one of the most fundamental mathematical inequalities. One reason for that is that in fact a great number of classical inequalities can be derived from (.), see e.g. [] and the references given therein. The inequality (.) cannot in general be improved since we have equality in (.) when φ(x)≡ x. However, f...

Asymptotic behavior of alternative Jensen and Jensen type functional equations

In 1941 D.H. Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 D.G. Bourgin was the second author to treat the Ulam problem for additive mappings. In 1982–2005 we established the Hyers–Ulam stability for the Ulam problem of linear and nonlinear mappings. In 1998 S.-M. Jung and in 2002–2005 the authors of this paper investigated the Hyers–Ulam stability of additive ...

On Some Improvements of the Jensen Inequality with Some Applications