Analysis of P-Superquadraticity and Related Integer and Fractional Order Inequalities with Applications

In this manuscript, we consider the concept of {P-} superquadratic functions and explore their key properties. From these properties, we can establish Jensen's, Mercer-Jensen's, and Hermite-Hadamard (H.H) type inequalities for this class of functions. The results are confirmed by the numerical computations and graphical illustration with several examples. Besides, a comparison is performed between the inequalities obtained through P-superquadratic functions and the ones acquired through P-convex functions. The analysis reflects that inequalities associated with P-superquadratic functions are far more refined than those deduced from P-convex functions. Additionally, we generalize the work with some fractional versions of the inequalities in question of H.H-type inequality involving Riemann-Liouville (R.L) fractional integral operators of P-superquadratic functions. We present several applications in the study involving type-1 modified Bessel functions, moments of random variables and special means. The novel results provided here not only advance the literature but significantly add new perspectives to it.