Jensen type inequalities for twice dierentiable functions

Abstract In this paper, we give some Jensen-type inequalities for ϕ : I → R, I = [α, β] ⊂ R, where ϕ is a continuous function on I, twice differentiable on ˚I = (α, β) and there exists m = inf ϕ// (x) or M = supϕ// (x) . x ˚I x ˚I Furthermore, if ϕ// is bounded on ˚I, then we give an estimate, from below and from above of Jensen inequalities.Qc 2012 NGA. All rights reserved.