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

In this paper we derive several improved forms of the Jensen inequality, giving the necessary and sufficient conditions for them to hold in the case of the real Stieltjes measure not necessarily positive. The obtained relations are characterized via the Green function. | Turk J Math (2017) 41: 1482 – 1496 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1610-26 Research Article More accurate Jensen-type inequalities for signed measures characterized via Green function and applications 1 ´ 1,∗, Josip PECARI ˇ ´ 2 , Mirna RODIC ´2 Mario KRNIC C Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb, Croatia 2 Faculty of Textile Technology, University of Zagreb, Zagreb, Croatia Received: 07.10.2016 • Accepted/Published Online: 26.01.2017 • Final Version: 23.11.2017 Abstract: In this paper we derive several improved forms of the Jensen inequality, giving the necessary and sufficient conditions for them to hold in the case of the real Stieltjes measure not necessarily positive. The obtained relations are characterized via the Green function. As an application, our main results are employed for constructing some classes of exponentially convex functions and some Cauchy-type means. Key words: Jensen inequality, Green function, refinement, convex function, exponentially convex function, Cauchy-type mean 1. Introduction The Jensen inequality is one of the most important inequalities in mathematical analysis and its applications. Recently, Peˇcari´c et al. [6] established conditions on a real Stieltjes measure dλ, not necessarily positive, under which the Jensen inequality and its reverse hold for a continuous convex function. These inequalities are characterized via the Green function G : [α, β] × [α, β] → R defined by { (t−β)(s−α) G(t, s) = for α ≤ s ≤ t, for t ≤ s ≤ β. β−α (s−β)(t−α) β−α (1) The corresponding result reads as follows: let g : [a, b] → [α, β] be a continuous function, and let λ : [a, b] → R be a continuous function or a function of a bounded variation such that λ(a) ̸= λ(b), and ∫b g(x)dλ(x) a∫ b dλ(x) a ∈ [α, β] . Then the following statements are equivalent: (i) For every continuous convex function φ : [α, β] → R the following .