Generalised Local Fractional Hermite-Hadamard Type Inequalities on Fractal Sets

Fractal geometry and analysis constitute a growing field, with numerous applications, based on the principles of fractional calculus. Fractals sets are highly effective in improving convex inequalities and their generalisations. In this paper, we establish a generalized notion of convexity. By defining generalised $\phi_{h-s}$ convex functions, we extend the well known concepts of generalised convex functions, $P$-functions, Breckner $s$-convex functions, $h$-convex functions amongst others. With this definition, we prove Hermite-Hadamard type inequalities for generalized $\phi_{h-s}$ convex mappings onto fractal sets. Our results are then applied to probability theory.


Introduction
Recent advancements have demonstrated the applications of fractional calculus in addressing real-life problems and enhancing a better understanding of intricate situations across various scientific fields.These applications span fields, including, but not limited to, probability models [38,27], physical modeling and experiments [47,6,7], control and dynamical systems [28,10,16], image processing [55,46], robotics and signal processing [18,58].Fractional calculus has contributed significantly to medical and biomedical research especially in neuroscience [48,24,21,52,43,44,59].Recent investigations extend to economic risk analysis [50], wind turbulence [19], and ongoing discussions across various disciplines.It has been used to improve the accuracy of several models in science and technology, finance and economics, medicine and engineering etc. Models of this nature have proven to exhibit greater efficiency compared to integer-order models.Most research in this direction have considered generalising existing theories and results to fractional order via fractional calculus.For instance, fractional Fourier transform [3], numerical and analytical fractional partial differential equations [39,49], L p theory and fractional order estimates [32], operator theory [26], fractional calculus of variations [2,8], generalised solutions to fractional div-curl systems [9] and more recently, their application to address challenges related to COVID-19 [40].For more details on fractional calculus see the books [37,30].
On the other hand, fractal geometry and analysis constitute a burgeoning area of study grounded in the principles of fractional calculus.Although the term "fractal" was introduced by Benoît Mandelbrot in 1975 in his memoir, later revised in 1982 [29], there is an extensive historical underpinning to fractals.Its rich history goes back to the time when mathematicians thought about certain curves and surfaces that differ conceptually from the classical ones in geometry.Fractals occur in nature in various forms and scales, exhibiting self-similar patterns at different magnifications.The geometry and analysis of fractals have been well fairly examined in literature.In a study on fractal dimension, [57] established some remarkable properties for the complement of a fractal set by relating its Besicovitch-Taylor index to an exterior dimension with specific applications in porous materials, blood network and the boundary of a diffusion process.Although fractals appears in several areas in mathematics and statistics such as geometric measure theory, topology, harmonic analysis, differential equations, numerical analysis, time series analysis [17,45,26,30,39,41], fractal mathematics is still yet to be fully explored.
Inequalities are basic building blocks in mathematics and other related fields.Several works focused on integral inequalities aim to extend, refine, improve, and generalize known inequalities, while also introducing new ones.
Fractional integral inequalities are useful tools in proving well-posedness and regularity of solutions to fractional partial differential equations [60].One of the most celebrated inequalities for convex functions is the Hermite-Hadamard inequality named after Charles Hermite and Jacques Hadamard.Let f : I ⊂ R → R be a convex function defined on an interval I, then for a, b ∈ I, 2 Generalised Form of Convexity on Fractal Set In [63], the author modified the concept of convex sets by introducing and developing the notion of E-convex sets, which we will refer to as the φ-convex sets in order to align with our notation.The author also defined φ-convex functions [63] and established some interesting results which have been applied to convex programming problems [5,14,42,54] with notable extensions on Riemannian manifolds [22,25].Since then, several definitions extending the concept of convexity within various contexts have been provided in literature.We recall the following definition from [35].
In general, φ h−s convex functions can be defined on φ-convex set in R n [63].We remark that with s = 1 and h(t) = 1 in Definition 2.1, for all s, t ∈ [0, 1], we recover Definition 3.1 in [63] and the results therein.Some properties of φ h−s convexity were established in [35] and several inequalities that generalise those of Jensen and Schur were derived for some non-negative supermultiplicative functions.Later, the class of harmonically φ h−s convex functions was given in [36] and applied to the Hermite-Hadamard inequality with several analytical implications for special means of real numbers.Also, in [36], the authors proved some Hermite-Hadamard-type inequalities by applying the fractional integral operator to φ h−s convex functions.The relationship between φ h−s convex functions and their harmonic counterparts was explored in [33], where several Pachpatte-type inequalities were established under a ratio-bound condition for the harmonically φ h−s convex functions.
Variations of Definition 2.1 have been used to establish Hermite-Hadamard-type inequalities on time scales [12,13].We adapt the above definition to fractal sets as follows.
Definition 2.2.Let I be an interval of R and let If the inequality (1) is reversed, then f is said to be generalised φ h−s concave, and of course f is generalised φ h−s convex if and only if −f is generalised φ h−s concave.If the inequality in (1) is strict whenever x and y are distinct and s, t ∈ [0, 1], then f is a strictly generalised φ h−s convex.
Remark 2.3.It is important to mention that the generalised class of φ h−s convex function given in Definition 2.2 unifies several interesting generalised versions of convex functions [11,61,31,53,56] some of which are highlighted below.
All the classes of convex functions listed above are generalised in the sense that they map intervals of R into fractal set.Throughout this work, we will use the function ρs : [0, ∞) → [0, ∞) defined by ρs(t) := .In most cases, we will restrict ρs on the interval [0, 1].There are several interesting properties of this new notion of generalised convex function some of which we shall now mention.To this end, the following remarks are in place.
which implies that f is also generalised φ h−s convex.An example is the function The Mittag-Leffler function [61,62] Γ(1+kα) for all x ∈ R is generalised convex [56], and hence by the remark above, a generalised φ h−s convex function provided ρs(t) ≥ t for all s, t ∈ [0, 1].
(ii) Consider the functions h1, h2 The converse is not true.
(iii) Let f, g : I → R α be two generalised φ h−s convex function and let λ α ∈ R α with λ > 0 in R. Then the functions f + g and λ α f are both generalised φ h−s convex functions on I.
(iv) Let f : I → R α be a generalised φ h−s convex function.If g : R → I is a linear map then the composition We state yet another property in the following Proposition whose proof is similar in spirit to that of Theorem 3.7 in [34] and Proposition 1 (and Corollary 1) in [36].
Proposition 2.5.Let f be a generalised φ h 1 −s convex function and g be a generalised φ h 2 −s convex function.Let h(t) := max{h1(t), h2(t)} with h(t) + h(1 − t) ≤ c for a fixed positive constant c.If f and g are similarly ordered, then f g is a generalised φ ch−s convex function.
Proof.Omitted.Proposition 2.6.Let fn : I → R α , n ∈ N, be a sequence of generalised φ h−s convex functions converging pointwise to a function f : I → R α .Then f is generalised convex on I.

Hermite-Hadamard-type Inequalities on Fractal Sets
We now establish some inequalities for generalised φ h−s convex functions on fractal sets.
Proof.Using the assumption of generalised φ h−s convexity on f , one observes that for all t ∈ [0, 1] and φ(a), φ(b Choosing , a local fractional integration of ( 5) with respect to t on [0, 1] yields the first inequality in (4).To prove the second inequality, we use the generalised φ h−s convexity of f to estimate the functional This completes the proof.Per Remark 2.3, several estimates can be deduced from Theorem 3.1.For instance, by choosing the functions φ(x) = x for all x ∈ I and h(t) = 1 for all t ∈ [0, 1], the following estimate involving Breckner s-convex functions in the second sense holds.Corollary 3.2.[31] Let f : I → R α .Suppose f ∈ Cα(I) is a generalised s-convex function in the second sense for s ∈ (0, 1).Then, for all t ∈ [0, 1] and a, b ∈ I ⊆ R with a = b, the following inequality holds Then, for all t ∈ [0, 1] and a, b ∈ I ⊆ R with a = b, the following inequality holds Corollary 3.4.Let f : I → R α .Suppose that f ∈ Cα(I) is a generalised P -function.Then, for all t ∈ [0, 1] and a, b ∈ I ⊆ R with a = b, the following inequality holds

Extensions of the Hermite-Hadamard Inequalities on Fractal Sets
We now establish an important representation lemma for functions belonging to the space Cα(I).