The classical Hermite–Hadamard inequality states that in any compact interval of \(\mathbb {R}\) , 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 \(f\in L[a,b]\) is a convex function, then for any \(p\in (a,b)\) the following inequalities are satisfied \(\begin{aligned}\begin{aligned} f\bigg (\dfrac{a+2p+b}{4}\bigg )\le \dfrac{1}{2}\bigg (\dfrac{1}{p-a}\int _{a}^p f(z)\,dz+\dfrac{1}{b-p}\int _{p}^b f(z)\,dz\bigg )\le \dfrac{f(a)+2f(p)+f(b)}{4}, \end{aligned} \end{aligned}\) \(\begin{aligned} \text{ and } \end{aligned}\) \(\begin{aligned}\begin{aligned} \begin{aligned}&\sup _{p\in (a,b)}\Bigg [\min \bigg \{f\bigg (\dfrac{a+p}{2}\bigg ),\,\,f\bigg (\dfrac{p+b}{2}\bigg )\bigg \}\Bigg ] \le \dfrac{1}{b-a}\int _{a}^{b} f(z)\,dz\\&\qquad \le \inf _{p\in (a,b)}\Bigg [\max \bigg \{\dfrac{f(a)+f(p)}{2},\,\,\dfrac{f(p)+f(b)}{2}\bigg \}\Bigg ]. \end{aligned} \end{aligned}\end{aligned}\) Alongside these results, for given non-negative convex functions \(f_1,\ldots , f_n\in L[a,b]\) , we derive a Hermite–Hadamard-type inequality for their product, incorporating the Beta function. We provide corollaries and comparative results to illustrate the generalizations and effectiveness of our findings. The research background, description of various notions, terminologies, and other crucial details can be found in the Introduction section.
