Generalized fractal Jensen and Jensen–Mercer inequalities for harmonic convex function with applications

In this paper, we present a generalized Jensen-type inequality for generalized harmonically convex function on the fractal sets, and a generalized Jensen–Mercer inequality involving local fractional integrals is obtained. Moreover, we establish some generalized Jensen–Mercer-type local fractional integral inequalities for harmonically convex function. Also, we obtain some generalized related results using these inequalities on the fractal space. Finally, we give applications of generalized means and probability density function.


Introduction
In different fields of pure and applied mathematics, the convexity of functions has been used. Several new classes of convex functions and convex sets have been introduced and studied. Many researchers have derived the variety of new inequalities associated with these new classes of convex functions [1][2][3]. The harmonic set was introduced by Shi et al. [4]. It must be noted that the weighted harmonic mean is used to give a concept of the harmonic set. It has applications in the theory of electrical circuits and other fields of sciences. Harmonic convex functions are defined by using the weighted harmonic means, which have appeared as a significant and major generalization of convex functions. Several properties of harmonic convex functions have been investigated by Íşcan [5], Dragomir [6], and Farid et al. in [7].
The definition of convex function is as follows.
Jensen's inequality (J. I) is the best-known result in the literature. The generalizations and improvements of Jensen's inequality have been a topic of supreme interest for researchers during the last few decades as evident from a large number of publications on the topic (see [8][9][10]). This inequality has also been used in various areas of science and technology to solve several problems, such as engineering, mathematical statistics, financial economics, and computer science. For some recent results, see [11][12][13]. In [14], Mercer gave a new variant of (J. I) as follows. In [5], Iscan gave the definition of harmonic convexity as follows.
Very recently, Baloch et al. [16] presented a variant of (J. I) in the Mercer sense for ∈ HK(I) as follows.
Fractional calculus and local fractional calculus are very powerful tools in applied mathematics [17][18][19][20][21]. Yang [22] stated the definition of local fractional calculus. The local fractional calculus is used to deal with various non-differentiable problems that arise in a complex real-world phenomenon system. Local fractional functional analysis is a fully new area of mathematics and also a totally new view of the mathematical world. The theory of fractional calculus has played an important role in various fields of applied and pure sciences, for example, electricity, mechanics, biology, economics, chemistry, notably control theory, image processing, etc. The local fractional calculus is extremely practical and comprehensive in science and engineering for its real-world models (see [23]). Mandelbrot defined a fractal set as the one whose Hausdorff dimension strictly exceeds the topological dimension [24]. Many researchers contemplated the properties of a function on the fractal space and built numerous sorts of fractional calculus by utilizing distinctive approaches, see [25,26]. Mo et al. [27] defined the generalized convex function on fractal sets R (0 < ≤ 1) of real numbers and established generalized Jensen's and Hermite-Hadamard's inequalities for a generalized convex function in the concept of local fractional calculus. In (2017) Sun [28] introduced the concept of harmonic convex function on fractal sets R (0 < ≤ 1) of real numbers and gave some Hermite-Hadamard inequalities for a generalized harmonic function ( ∈ GHK (I)). In 2018, authors [29] worked on fractal integral inequalities for ∈ GHK(I). Sun [30] in 2019 worked on generalization of some inequalities for ( ∈ GHK (I)) on the fractal space. Recently, in 2020, Iftikhar et al. [31] gave several new Newton-type inequalities in a local fractional calculus scheme. Some recent results for a generalized harmonic convex function can be seen in [32].

Preliminaries
Using the Gao-Yang-Kang concept, recall the set R to classify the definitions of local fractional derivative, local fractional integral (see [22]), and so on.
The theory of Yang's fractional sets [22] can be stated as follows. For 0 < ≤ 1, the -type set of element set is given below: Z : The set defined as the -type set of integer is {0 , ±1 , ±2 , . . . , ±n , . . .}. Q : The set defined as the -type set of rational numbers is {m = (r/s) : r, s ∈ Z, s = 0}. J : The set defined as the -type set of irrational numbers is {m = (r/s) : r, s ∈ Z, s = 0}. R : The set defined as the -type set of real numbers is R = Q ∪ J . The following operations hold for r , s , and t belonging to the set R of real line numbers: (i) r + s and r s belong to the set R ; (ii) r + s = s + r = (r + s) = (s + r) ; (iii) r + (s + t ) = (r + s) + t ; (iv) r s = s r = (rs) = (sr) ; (v) r (s t ) = (r s )t ; (vi) r (s + t ) = r s + r t ; (vii) r + 0 = 0 + r = r and r 1 = 1 r = r . Let us recall some basics of local fractional calculus on R .

Definition 4 ([22])
The definition of local fractional derivative of (ζ ) of order at ζ = ζ 0 is Here, it implies that m I ( ) The definition of a generalized harmonically convex function on fractal sets is as follows.
The main aim of this paper is to present the generalized Jensen inequality (G. J. I) and generalized Jensen-Mercer inequality (G. J. M. I) for the class of functions GHK (I) on the fractal space. Moreover, we establish an improvement and generalization of some Jensen-Mercer-type inequalities for harmonically convex function via local fractional integrals. Also, we obtain some generalized related results on a fractal space. Finally, we present some resulting applications to special means and probability density function.

Generalized Jensen's and Jensen-Mercer inequalities for GHK (I) in the fractal sense
In this section, we first present the (G. J. I) and establish (G. J. M. I) for ∈ GHK (I) via fractional integrals. In order to prove (G. J. M. I), we need the main identity later in this section.
Remark 1 If we take = 1 in Theorem 5, then it gives inequality (1) proved by Dragomir.
The main lemma for ∈ GHK (I) pertaining local fractional integrals is as follows.
M so that the pairs m, M and a ı , b ı possess the same harmonic mean. Since that is the case, there exists ϑ such that where 0 ≤ ϑ ≤ 1 and 1 ≤ ı ≤ s. Hence, applying generalized harmonic convexity of , we get

Improvement and generalization of some (G. J. M)-type inequalities for local fractional integrals
Based on Lemma 3, some (G. J. M)-type inequalities can be represented pertaining local fractional integral forms as follows.

Related results
In this section, we present the following related results for local fractional integrals.
Remark 8 If we take = 1 in Theorem 10, then it reduces to Theorem 3.4 (see [16]). Proof that is, and so the first inequality of (19) is proved. If ∈ GHK (I), then for ϑ ∈ [0, 1] we have Multiplying by 1 (1+ ) on both sides of the above equation and then integrating w.r.t ϑ over [0, 1], we get Adding (M) + (m) to both sides of the above equation, we have Combining (20) and (21), we get the second inequality of (19).

Conclusion
In this paper, for the first time, we introduce Jensen's inequality for harmonic convex functions by using local fractional calculus. As a result we also introduce a variant of generalized Jensen-Mercer inequality in the fractal sense. Then we present the main lemma involving local fractional integrals. By using this main lemma, we establish generalized Jensen-Mercer-type inequalities on fractal sets R (0 < ≤ 1). Moreover, an improvement and generalization of some results related to the class of generalized harmonically convex function via local fractional integrals are established. Using these inequalities, we obtain some generalized related results on the fractal space. Lastly, some applications to some special means of real numbers and probability density function are established. It is quite open to investigate Jensen and Jensen-type inequalities for other classes of generalized convex functions in the fractal sense. Since there is a massive body of literature about refinements, converses, and reverses of such Jensen and Jensen-type inequalities, so their generalized variants using local fractional calculus can be revealed also. One of the important directions is to give fractal Jensen-Mercer-Ostrowski type inequalities and give applications to continuous random variables and improve quadrature rules. Since local fractional calculus is quite effective from an application point of view, a bridge between inequality theory and their applications can be established, which may eventually generate optimal solutions.