New Hermite-Hadamard type inequalities for exponentially convex functions and applications

The investigation of the proposed techniques is effective and convenient for solving the integrodifferential and difference equations. The present investigation depends on two highlights; the novel Hermite-Hadamard type inequalities for $\mathcal{K}$-conformable fractional integral operator in terms of a new parameter $\mathcal{K}>0$ and weighted version of Hermite-Hadamard type inequalities for exponentially convex functions in the classical sense. By using an integral identity together with the Holder-Iscan and improved power-mean inequality we establish several new inequalities for differentiable exponentially convex functions. This generalizes the Hadamard fractional integrals and Riemann-Liouville into a single form. Our contribution expands some innovative studies in this line. Moreover, two suitable examples are presented to demonstrate the novelty of the results established, the first one about the contributions of the modified Bessel functions and the other is about $\sigma$-digamma function. Finally, various applications for some special means as arithmetic, geometric and logarithmic are given.