Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications

The present article addresses the concept of p-convex functions on fractal sets. We are able to prove a novel auxiliary result. In the application aspect, the fidelity of the local fractional is used to establish the generalization of Simpson-type inequalities for the class of functions whose local fractional derivatives in absolute values at certain powers are p-convex. The method we present is an alternative in showing the classical variants associated with generalized p-convex functions. Some parts of our results cover the classical convex functions and classical harmonically convex functions. Some novel applications in random variables, cumulative distribution functions and generalized bivariate means are obtained to ensure the correctness of the present results. The present approach is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractals in computer graphics.

With the development of the inequality theory, the inequalities for various kinds of convex functions have a rapid blossom in the area of pure and applied mathematics [30][31][32][33][34][35][36][37][38][39][40][41][42]. As mentioned above, many articles are all involved with Hermite-Hadamard type and trapezoid type, midpoint type inequalities [43][44][45][46][47][48][49]. However, to the best of our knowledge, Simpson-type inequalities for functions whose first localα-derivatives in absolute value are the class of generalized p-convex functions have not been reported. So we turn our attention to this new research.
One of the aspects which are nowadays particularly well known among researchers is the integral inequalities with applications. In this field, the majority of the authors are generalizing the standard results in the accessible literature by utilizing various sorts/definitions of the fractional integral operators [4,25,50,51]. An enormous heft of fractional differential problems and partial differential equations can be converted into problems of comprehending some estimated integral equations.
The early research motivations in the area of the local fractional theory were for solving a bulk of initial and boundary value associated with differential equations [52,53]. In [53], Yang introduced a contemporary study used to tackle nondifferentiable problems that incorporate in complex systems of real-world phenomena. The fractal sets in science have introduced some fascinating complex graphs and picture compressions to computer graphics. The expression "fractal" was first utilized by a young mathematician, Julia [54] when he was considering Cayley's problem identified with the conduct of Newton's method in a complex plane. The fractal is frequently utilized in real-world studies, involving fractal antennas, fractal transistors, and fractal heat exchangers. It has applications in the music industry, the creation of photography, soil mechanics, small-angle scattering theory, and many more fileds. It is to be emphasized that fractal theory assumes an essential job in the improvement of picturing of fractal sets. The utilizations of fractal sets in cryptography and other useful areas of research have increased the interest of researchers to broaden the utilization in mathematical inequalities. Fractals are particular arbitrary examples descriptions addressing the erratic developments of the disorderly world at work. The most significant utilization of fractals in software engineering is fractal picture compression. This sort of compression utilizes the way that this present reality is very well portrayed by fractal geometry [52,55,56]. Interestingly, the author of [53] investigated the local fractional functions on fractal space deliberately, which comprises of local fractional calculus and the monotonicity of functions. Numerous analysts contemplated the characteristics of functions on fractal space and built numerous sorts of fractional calculus by utilizing various strategies [57][58][59]. Additionally, integral inequalities in the context of local fractional calculus have a significant role in all fields of pure and applied mathematics. For example, Chen [60] derived a novel version of Hölder inequality on fractals. In [58], Mo [62], deduced several new Fejér-Hermite-Hadamard inequalities for a class of h-convex functions with applications. For some useful and recent studies on fractional calculus and its applications in different fields of mathematics [63][64][65][66][67][68].
Owing to the above phenomena, the key aim of this research is to introduce a new auxiliary result depending on local fractal sets are presented. With the aid of novel identity, we derived numerous novel generalizations of Simpson-type for mappings whose powers contain local fractional derivatives in modulus are generalized p-convex. The main impetus of this study to capture new estimates for generalized convex functions and generalized harmonically convex functions. In addition, the application of the proved results in a random variable, cumulative distribution functions, and the generalized bivariate mean formula is also presented. We hope that the new strategy formulated in the present paper is more invigorating than the accessible one.

Preliminaries
Now, we mention the preliminaries from the theory of local fractional calculus. These ideas and important consequences associated with the local fractional derivative and local fractional integral are mainly due to Yang et al. [53].
We clearly see that

Main results and discussions
In this section, we first inaugurate a local fractional integral identity for generalized pconvex functions, and then we utilize the said identity to establish certain Simpson-type variants in the context of the fractal domain. We now present the concept of generalized p-convex functions on fractal space as follows.
Remark 3.2 From Definition 3.1 we clearly see that: (1) If we takeα = 1, then we get a definition given in [69].
(2) If we take p = 1, then we get a definition given in [58].
(3) If we take p = -1, then we get a definition given in [66].
(4) If we take p = 1 withα = 1, then we get the classical convex function.
It is worth mentioning that generalized p-convex functions collapse to generalized convex, generalized harmonically convex functions, harmonically convex functions, and classical convex functions as special cases. This shows that outcomes derived in the present paper continue to hold for these classes of convex functions and their variant forms.
. Then the inequality Proof Firstly, let us calculate the following integrals: Utilizing the local fractional integration by parts and the change of the variable technique for the integrals I 1 and I 2 , we get and Adding (3.2) and (3.3) and then multiplying the obtained result by 1 2α , we have Equation (3.4) gives the desired result.
and |χ (α) | q is a generalized p-convex function on . Then one has (

3.11)
Proof It follows from Lemma 3.4 and the generalized power mean inequality that Making use of the generalized p-convexity of |χ (α) | q on , we have We use Lemma 2.5 and the facts that where we have used the identities  (1) Letα = 1. Then we get Theorem 2.1 of [29].