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In this paper, some new fractal versions of Fejér–Hermite–Hadamard (FHH) type variants for generalized Raina (Ψ , ℏ) -convex mappings are established benefiting from Raina's function and fractal set ℝ φ , 0 < φ < 1. By means of three integral identities coupled with Raina's function and local differentiation, we established some bounds for the difference between the left and central parts and also the difference between the center and right parts in FHH inequality. Besides that, some illustrative examples and noted special cases are apprehended. Additionally, we developed various generalizations for random variables, cumulative distribution functions, and special function theory as applications of local fractional integrals. The consequences established can provide contribution to inequality theory, fractional calculus and probability theory from the viewpoint of application to establish the other associated classes of functions. With the aid of these methodologies, it is promising to comprise further bounds of other type of variants which involve local fractional techniques. [ABSTRACT FROM AUTHOR]

In the present study, two new classes of convex functions are established with the aid of Raina’s function, which is known as the ψ-s-convex and ψ-quasi-convex functions. As a result, some refinements of the Hermite–Hadamard (ℋℋ{\mathcal{ {\mathcal H} {\mathcal H} }})-type inequalities regarding our proposed technique are derived via generalized ψ-quasi-convex and generalized ψ-s-convex functions. Considering an identity, several new inequalities connected to the ℋℋ{\mathcal{ {\mathcal H} {\mathcal H} }} type for twice differentiable functions for the aforesaid classes are derived. The consequences elaborated here, being very broad, are figured out to be dedicated to recapturing some known results. Appropriate links of the numerous outcomes apprehended here with those connecting comparatively with classical quasi-convex functions are also specified. Finally, the proposed study also allows the description of a process analogous to the initial and final condition description used by quantum mechanics and special relativity theory.

Abstract We establish certain new fractional integral inequalities involving the Raina function for monotonicity of functions that are used with some traditional and forthright inequalities. Taking into consideration the generalized fractional integral with respect to a monotone function, we derive the Grüss and certain other associated variants by using well-known integral inequalities such as Young, Lah–Ribarič, and Jensen integral inequalities. In the concluding section, we present several special cases of fractional integral inequalities involving generalized Riemann–Liouville, k-fractional, Hadamard fractional, Katugampola fractional, ( k , s ) $(k,s)$ -fractional, and Riemann–Liouville-type fractional integral operators. Moreover, we also propose their pertinence with other related known outcomes.

We establish certain new fractional integral inequalities involving the Raina function for monotonicity of functions that are used with some traditional and forthright inequalities. Taking into consideration the generalized fractional integral with respect to a monotone function, we derive the Grüss and certain other associated variants by using well-known integral inequalities such as Young, Lah–Ribarič, and Jensen integral inequalities. In the concluding section, we present several special cases of fractional integral inequalities involving generalized Riemann–Liouville, k-fractional, Hadamard fractional, Katugampola fractional, (k , s) -fractional, and Riemann–Liouville-type fractional integral operators. Moreover, we also propose their pertinence with other related known outcomes. [ABSTRACT FROM AUTHOR]