Reverses of the Jensen-Type Inequalities for Signed Measures

Abstract

In this paper we derive refinements of the Jensen type inequalities in the case of real Stieltjes measuredλ, not necessarily positive, which are generalizations of Jensen's inequality and its reverses for positive measures. Furthermore, we investigate the exponential and logarithmic convexity of the difference between the left-hand and the right-hand side of these inequalities and give several examples of the families of functions for which the obtained results can be applied. The outcome is a new class of Cauchy-type means.

Some Generalizations of the Jensen-Type Inequalities with Applications

“…Taking into account the properties of the functions G p (p = 1, 2, 3, 4), and motivated by the results from S. S. Dragomir in [16,17] with reverses of the Jensen inequality in the case when the measure is positive, here in this paper we will give the generalization of some of his results on the Jensen-type inequalities and the converse Jensen-type inequalities, but now allowing that the measure can also be negative. The results that are presented here, represent the continuation of the research presented in [18,19].…”

“…If φ is convex function, then for every s ∈ [α, β] we have that φ (s) ≥ 0. Further, if for every s ∈ [α, β] holds (17), then the term in the square brackets in (18) is less then or equal to zero. This leads to conclusion that also the left hand side of (18) has to be less than or equal to zero, i.e., that for every contionuos convex function φ : (16).…”

More accurate Jensen-type inequalities for signed measures characterized via Green function and applications

“…In this section we give improved versions of the Jensen inequality (2) and the Lah-Ribarič inequality (4); that is, we give the corresponding refinements and reverses. As we have already mentioned, these more accurate relations are also characterized by the Green formula.…”

“…It should be noticed here that in the previous theorem g does not have to belong to the interval [α, β]. In the case when g ∈ [m, M ], the inequality (14) represents the reverse of the Lah-Ribarič inequality(4), while the inequality with the reversed sign represents its refinement. Theorems 2.5 and 2.8 refer to a convex function φ.…”

“…Motivated by [6], Jakšić et al [4] obtained several reverses of (2) also characterized via the Green function. In particular, they showed that if (5) holds, then for every continuous convex function φ : [α, β] → R the following inequality holds:…”