Jensen type inequalities for twice dierentiable functions

Abstract

In this paper, we give some Jensen-type inequalities for \(\varphi: I\rightarrow\mathbb{R}, I=[\alpha,\beta ]\subset\mathbb{R}\) where \(\varphi\) is a continuous function on \(I\); twice differentiable on \(I^°=(\alpha,\beta )\) and there exists \(m = \inf _{x\in I^°} \varphi ''(x)\) or \(M = \sup_{x\in I^°}\varphi '' (x)\). Furthermore, if \(\varphi ''\) is bounded on \(I^°\) ; then we give an estimate, from below and from above of Jensen inequalities.