Abstract
We observe that the Hermite–Hadamard inequality written in the form $$f\left(\frac{x+y}{2}\right)\leq\frac{F(y)-F(x)}{y-x}\leq\frac{f(x)+f(y)}{2}$$ may be viewed as an inequality between two quadrature operators \({f\left(\frac{x+y}{2}\right)}\)\({\frac{f(x)+f(y)}{2}}\) and a differentiation formula \({\frac{F(y)-F(x)}{y-x}}\). We extend this inequality, replacing the middle term by more complicated ones. As it turns out in some cases it suffices to use Ohlin lemma as it was done in a recent paper (Rajba, Math Inequal Appl 17(2):557–571, 2014) however to get more interesting result some more general tool must be used. To this end we use Levin–Steckin theorem which provides necessary and sufficient conditions under which inequalities of the type we consider are satisfied.
Highlights
We observe that the Hermite–Hadamard inequality written in the form f x+y 2
We extend this inequality, replacing the middle term by more complicated ones. As it turns out in some cases it suffices to use Ohlin lemma as it was done in a recent paper (Rajba, Math Inequal Appl 17(2):557–571, 2014) to get more interesting result some more general tool must be used
We shall obtain some class of inequalities of the Hermite–Hadamard type
Summary
We shall obtain some class of inequalities of the Hermite–Hadamard type. First we write the classical Hermite–Hadamard inequality f y1 −x y f (t)dt ≤ f (x) + f (y) x (1)A. [7] Ohlin lemma on convex stochastic ordering was used to obtain inequalities of the Hermite-Hadamard type. We shall use Theorem 1 to make an observation which is more general than Ohlin lemma and concerns the situation when functions F1, F2 have more crossing points than one.
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