In this paper, we have established the Hermite–Hadamard–Fejér inequality for fractional integrals involving preinvex functions. The results presented here provide new extensions of those given in earlier works as the weighted estimates of the left and right hand side of the Hermite–Hadamard inequalities for fractional integrals involving preinvex functions doesn’t exist previously.

In this article, we have established some new bounds of Fejér-type Hermite-Hadamard inequality for k -fractional integrals involving r -times differentiable preinvex functions. It is noteworthy that in the past, there was no weighted version of the left and right sides of the Hermite-Hadamard inequality for k -fractional integrals for generalized convex functions available in the literature.

There is a strong correlation between the concept of convexity and symmetry. One of these is the class of interval-valued cr-log-h-convex functions, which is closely related to the theory of symmetry. In this paper, we obtain Hermite–Hadamard and its weighted version inequalities that are related to interval-valued cr-log-h-convex functions, and some known results are recaptured. To support our main results, we offer three examples and two applications related to modified Bessel functions and special means as well.

The unified Einstein–Euler–Heisenberg theory is utilized to investigate the particle motion and weak gravitational lensing characteristics of black holes. This black hole solution is developed using spherically symmetric possessing electric and magnetic charges. Quantum electrodynamics corrections reveal a screening effect for BH electric charges and paramagnetic impacts on magnetic charges. We analyzed the motion of massive as well as massless particles by studying their effective potential, event horizon, photon orbit and inner circular orbit. It was demonstrated that magnetic and electric fields of spherically symmetric black holes have significant impact. Then, we also delve to study the weak gravitational lensing phenomenon. A comprehensive approach was employed to investigate this phenomenon and explore the angle of deflection of light rays near magnetically and electrically charged black holes.