On generalization, refinement, and extension of Hermite–Hadamard inequality

Abstract The classical Hermite–Hadamard inequality states that in any compact interval of $$\mathbb {R}$$ R , the integral mean of a convex function is greater than the functional value at the midpoint, while it is bounded above by the average of the functional values at the endpoints. This paper presents some generalizations, refinements, and extensions of the Hermite–Hadamard inequality for convex functions. We show that if $$f\in L[a,b]$$ f ∈ L [ a , b ] is a convex function, then for any $$p\in (a,b)$$ p ∈ ( a , b ) the following inequalities are satisfied $$\begin{aligned}\begin{aligned} f\bigg (\dfrac{a+2p+b}{4}\bigg )\le \dfrac{1}{2}\bigg (\dfrac{1}{p-a}\int _{a}^p f(z)\,dz+\dfrac{1}{b-p}\int _{p}^b f(z)\,dz\bigg )\le \dfrac{f(a)+2f(p)+f(b)}{4}, \end{aligned} \end{aligned}$$ f ( a + 2 p + b 4 ) ≤ 1 2 ( 1 p - a ∫ a p f ( z ) d z + 1 b - p ∫ p b f ( z ) d z ) ≤ f ( a ) + 2 f ( p ) + f ( b