Caputo–Fabrizio fractional Hermite–Hadamard type and associated results for strongly convex functions

Abstract

The study of fractional integral inequalities has attracted the interests of many researchers due to their potential applications in various fields. Estimates obtained via strongly convex functions produce better and sharper bounds when compared to convex functions. To this end, we establish some new Hermite–Hadamard and Fejér types inequalities by means of the Caputo–Fabrizio fractional integral operators for strongly convex functions. In particular, we prove among other things that if $\omega :\mathfrak {I}\rightarrow {\mathbb {R}}$ is a strongly convex function with modulus $c>0$ and $\alpha ,\beta \in \mathfrak {I}$ with $\alpha <\beta$, then

$$\begin{aligned} \begin{aligned}&\omega \left( \frac{\alpha +\beta }{2}\right) +\frac{c}{12}(\beta -\alpha )^2\\&\quad \le \frac{B(\mu )}{\mu (\beta -\alpha )}\left[ {^{cf}{\mathcal {I}}}_{\alpha }^{\mu }\omega (s)+^{cf}{\mathcal {I}}_{\beta }^{\mu }\omega (s)-\frac{2(1-\mu )}{B(\mu )}\omega (s)\right] \\&\quad \le \frac{\omega (\alpha )+\omega (\beta )}{2}-\frac{c}{6}(\beta -\alpha )^2, \end{aligned} \end{aligned}$$

where $\mu \in (0,1]$, $s\in \mathfrak {I}$ and $B(\mu )>0$ is a normalization function. Some applications to special means have also been investigated.