Generalized trapezium-type inequalities in the settings of fractal sets for functions having generalized convexity property

In the paper, we extend some previous results dealing with the Hermite–Hadamard inequalities with fractal sets and several auxiliary results that vary with local fractional derivatives introduced in the recent literature. We provide new generalizations for the third-order differentiability by employing the local fractional technique for functions whose local fractional derivatives in the absolute values are generalized convex and obtain several bounds and new results applicable to convex functions by using the generalized Hölder and power-mean inequalities. As an application, numerous novel cases can be obtained from our outcomes. To ensure the feasibility of the proposed method, we present two examples to verify the method. It should be pointed out that the investigation of our findings in fractal analysis and inequality theory is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.


Introduction
Fractals are mathematical developments that present self-similarity over a scope of scales and noninteger (fractal) measurements. Attributable to these properties, Yang [1] contemplated a new investigation to tackle some nondifferentiable problems that can be utilized to productively assess the geometrical multifaceted nature, and the anomaly of shapes and examples seen in human disease through the utilization of customary Euclidean geometry in such figurings is all the more challenging [2,3]. The use of fractal analysis in image processing, machine learning, cryptography, electrochemical processes, physics, diagnostic imagining, neuroscience, image analysis, acoustic, physiology, and Riemann zeta zeros has shown considerable guarantee for estimating forms that have changed as ordinarypartial differential equations [4][5][6][7][8]. Regardless of the benefits of fractal mathematics and various examinations exhibiting its pertinence to porous media, aquifer, turbulence, and more other media commonly displaying fractal properties, numerous specialists and researchers stay uninformed of its latent capacity involving local fractional articulations.
Moreover, local fractional derivatives and integrals are viably applied numerous conditions, for instance, the Fokker-Planck equation, the diffusion and relaxation equation in fractal space, the fractal heat conduction equation, and the local fractional diffusion equation [9]. Moreover, fractional calculus (integrals and derivatives) has gained significant interest throughout the past three decades due chiefly to its incontestable utilities in numerous areas of science and technology [10,11]. The fractional operator does indeed provide many potentially helpful tools for numerous problems involving special functions of mathematical science and their extensions and generalizations in one and several variables. Fractional integrals are utilized for depiction of various hereditary and memory effects of different processes and constituents in physical processes like seepage flow in fluid dynamic traffic model and nonlinear oscillations of earthquake [12,13].
Fractional integrals have been analyzed for integral inequalities and solution of fractional differential and difference equations. The Hermite-Hadamard inequality [14] is one of such type inequalities, extensively used in the literature and providing a necessary and sufficient condition for a function to be convex. Now we recall the Hermite-Hadamard inequality. Let G : ⊆ R → R be a convex function. Then we have the double inequality for all η 1 , η 2 ∈ with η 1 = η 2 .
Recently, generalizations, extensions, improvements, and variants of the Hermite-Hadamard inequality have attracted the attention of many researchers due to its wide applications in pure and applied mathematics. The main purpose of the paper is presenting the fundamental basis of fractals and illustrating analysis of fractal sets and related estimations, in particular, establishing the integral inequalities for the functions whose local fractional differentiation in the absolute values are generalized convex.
Local fractional inequalities and their fertile applications in pure and applied mathematics have attracted the attention of many researchers [15,16]. For example, Mo et al. [17] derived the generalized Hermite-Hadamard inequality for generalized convex functions.
is the Mittag-Leffler function. Recently, the fractal theory has achieved a considerable interest. Mo et al. [17] found the following analogue of the Hermite-Hadamard inequality (1.1) for generalized convex functions:

Identity via first-order local differentiable mappings
To establish our main results, we need the following lemma.
Proof By using local fractional integration by parts and change of variable we have By adding identities (3.2) and (3.3) we get the desired identity (3.1).

Hermite-Hadamard-type inequalities for first-order differentiable functions
Using Lemma 3.1, we now give some novel generalizations of the Hermite-Hadamardtype inequality for functions with generalized convex local derivatives.
Proof It follows from Lemma 3.1 and the modulus property that where we have used the equalities This completes the proof.
Some particular remarkable cases of Theorem 4.1 are as follows.
Some particular cases of Theorem 4.5 and corollaries are as follows.
Proof It follows from Lemma 3.1 and the generalized power-mean inequality that Since |G (α) | ρ is a generalized convex function, we obtain Similarly, Combining (4.6)-(4.8) gives the desired inequality (4.5). This completes the proof.

Corollary 4.8
Under the assumptions of Theorem 4.7, if we take y = η 1 +η 2 2 , then we have

Generalized inequalities for second-order differentiability
This section is devoted to certain generalizations for twice locally differentiable functions, which are connected with the Hermite-Hadamard-type inequality. For this purpose, we need the following lemma.
Lemma 5.1 Letα ∈ (0, 1], and let a mapping G : → Rα be such that G ∈ Dα( ) and Then Proof Using local fractional integration by parts and change of variable, we have and, analogously, Adding (5.2) and (5.3) and then multiplying both sides by ( (η 2 -η 1 ) 2 16 )α give the desired identity. This completes the proof.

Theorem 5.3 Let a mapping
, and |G (2α) | is a generalized convex function on . Then Proof From Lemma 5.1 and the modulus property we have Since |G (2α) | is a generalized convex function on , we have and Therefore where we have used the equality This completes the proof.

Generalized integral inequalities for third-order local differentiable functions
In this section, we present some novel variants of Hermite-Hadamard-type inequality for the functions with generalized convex third local fractional derivatives. For this purpose, we need the following identity. Lemma 6.1 Letα ∈ (0, 1], and let a mapping G : → Rα be such that G ∈ Dα( ) and G (3α) ∈ Cα[η 1 , η 2 ]. Then Proof Using the local fractional integration by parts and change of variable leads to Analogously, we have Subtracting the above identities side by side and then multiplying the obtained identity by ( (η 2 -η 1 ) 3
We have some particular cases of Theorem 6.5.

Corollary 6.6
Under the suppositions of Theorem 6.5, we have the following conclusions.

Conclusion
In the paper, we investigated the local fractional differentiation and integration for the generalized convex functions. Under this approach, we have derived three identities related to many well-known inequalities in the literature. For generalized convex functions, we obtained several novel bounds for higher-order local differentiable functions in different forms, which lead to the bounds of several known results in [23,26]. With reference to the definition of generalized convex functions, there is much to explore in the area of fractal analysis and machine learning by introducing specific values of the fractal parameters. Our ideas and approach may lead to a lot of follow-up research.