On a new converse of Jensen’s inequality. (English) Zbl 1224.26036

Simić, Slavko

Publ. Inst. Math., Nouv. Sér. 85(99), 107-110 (2009).

The author presents a new upper bound for Jensen’s discrete inequality: If \(f\) is a real convex function defined on \([a,b]\), and \(x_i\in[a,b]\), \(p_i,p,q>0\), \(i=1,2\dots,n\), such that \(\sum_{i=1}^np_i=1\), \(p+q=1\), then \(\sum_{i=1}^n p_if(x_i)-f\left(\sum_{i=1}^np_ix_i\right)\leq\max_p\big[pf(a)+qf(b)-f(pa+qb)\big]\). It is proved that this bound is better than the existing ones, and some applications are given.

Reviewer: Slobodanka Janković (Beograd)

MSC:

26A51	Convexity of real functions in one variable, generalizations
26D15	Inequalities for sums, series and integrals

Keywords:

convex function