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In the paper, we introduce the generalized convex function on fractal sets
and study the properties of the generalized convex function. Based on these
properties, we establish the generalized Jensen's inequality and generalized
Hermite-Hadamard's inequality on fractal sets. Furthermore, some applications
are given .

To read the full-text of this research,

you can request a copy directly from the authors.

... Definition 9 (see [42]). A function ( ) is called -convex on if the following inequality holds true: ...

... Theorem 10 (see [42]). Assume that ( ) is an -local differentiable function on . ...

... Theorem 11 (see [42]). Assume that ( ) is a local fractional continuous function. ...

Anderson's inequality (Anderson, 1958) as well as its improved version given by Fink (2003) is known to provide interesting examples of integral inequalities. In this paper, we establish local fractional integral analogue of Anderson's inequality on fractal space under some suitable conditions. Moreover, we also show that the local fractional integral inequality on fractal space, which we have proved in this paper, is a new generalization of the classical Anderson's inequality.

... On the fractal space, Yang studies the monotonicity of function by the local fractional derivative [15]. And, in [16] the authors introduce the generalized convex function and study the analysis properties of the generalized convex function on fractal sets. The generalized convex function on fractal sets R α (0 < α < 1) can be stated as: f : [a, b] ⊂ R → R α is said to be generalized convex on [a, b], if inequality ...

... Note that, when s = 1, the generalized s-convex function in both sense is generalized convex function, see [16]. ...

In the paper, we introduce two kinds of generalized s-convex functions on
fractal sets. And similar to the class situation, we also study the properties
of these two kinds of generalized s-convex functions and discuss the
relationship between them. Based on these properties, we get some applications
on fractal sapce.

... 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 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.

... In [11], Mo et al. proved the following generalized Hermite-Hadamard inequality for generalized convex function: ...

First, we establish the generalized Ostrowski inequality for local fractional integrals on fractal sets Rα (0 < α ≤ 1) of real line numbers. Secondly, we obtain some new inequalities using the generalized convex function on fractal sets Rα.

... On the fractal set, Mo etal. [7,8] introduced the definition of the generalized convex function and established Hermite-Hadamard type inequality. In [9], the authors introduced two kinds of generalized s-convex functions on fractal sets R α (0 < α < 1). ...

In the paper, we establish the Hermite-Hadamard type inequalities for the
generalized s-convex functions in the second sense on real linear fractal set
$\mathbb{R}^{\alpha}(0<\alpha<1).$

Inequalities play important roles not only in mathematics but also in other fields, such as economics and engineering. Even though many results are published as Hermite–Hadamard (H-H)-type inequalities, new researchers to these fields often find it difficult to understand them. Thus, some important discoverers, such as the formulations of H-H-type inequalities of α-type real-valued convex functions, along with various classes of convexity through differentiable mappings and for fractional integrals, are presented. Some well-known examples from the previous literature are used as illustrations. In the many above-mentioned inequalities, the symmetrical behavior arises spontaneously.

This paper is devoted to establishing some functional generalizations of Hölder and reverse Hölder’s inequalities with local fractional integral introduced by Yang. Then, based on the obtained results, we derive some related inequalities including local fractional integral Minkowski-type and Dresher-type inequalities, which are some extensions of several existing local fractional integral inequalities.

In this paper, the Hermite-Hadamard inequality for s-convex functions in the third sense is provided. In addition, some integral inequalities for them are presented. Also, the new functions based on the integral and double integral of s-convex functions in the third sense are defined and under certain conditions, the third sense s-convexity of these functions are shown and some inequality relations for these are expressed.

In this article, we establish new Hermite–Hadamard-type inequalities via Riemann–Liouville integrals of a function ψ taking its value in a fractal subset of R and possessing an appropriate generalized s-convexity property. It is shown that these fractal inequalities give rise to a generalized s-convexity property of ψ . We also prove certain inequalities involving Riemann–Liouville integrals of a function ψ provided that the absolute value of the first or second order derivative of ψ possesses an appropriate fractal s-convexity property.

The purpose of this article is to present some new inequalities for products of generalized convex and generalized s-convex functions on fractal sets. Furthermore, some applications are given.
MSC: 26A51, 26D07, 26D15, 53C22.

Maxwell’s equations on Cantor sets are derived from the local fractional vector calculus. It is shown that Maxwell’s equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Local fractional differential forms of Maxwell’s equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are obtained. Maxwell's equations on Cantor set with local fractional operators are the first step towards a unified theory of Maxwell’s equations for the dynamics of cold dark matter.

In this article the local fractional Schrödinger equations in the one-dimensional Cantorian system are investigated. The approximations solutions are obtained by using the local fractional series expansion method. The obtained solutions show the present method is an efficient and simple tool for solving the linear partial differentiable equations within the local fractional derivative.

Yang-Fourier transform is the generalization of the fractional Fourier transform of non-differential functions on fractal space. In this paper, we show applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral

Recently, new notions such as local fractional derivatives and local fractional differential equations were introduced. Here we argue that these developments provide a possible calculus to deal with phenomena in fractal space-time. We show how the usual calculus is generalized to deal with non Lipschitz functions. We also indicate how a definition of a fractal measure arises from these developments much the same way as the Lebesgue measure from ordinary calculus.

The framework for the mechanics of solids, deformable over fractal subsets, is outlined. While displacements and total energy maintain their canonical physical dimensions, renormalization group theory permits to define anomalous mechanical quantities with fractal dimensions, i.e., the fractal stress [σ*] and the fractal strain [ε*]. A fundamental relation among the dimensions of these quantities and the Hausdorff dimension of the deformable subset is obtained. New mathematical operators are introduced to handle these quantities. In particular, classical fractional calculus fails to this purpose, whereas the recently proposed local fractional operators appear particularly suitable. The static and kinematic equations for fractal bodies are obtained, and the duality principle is shown to hold. Finally, an extension of the Gauss–Green theorem to fractional operators is proposed, which permits to demonstrate the Principle of Virtual Work for fractal media.

Many biologists now recognize that environmental variance can exert important effects on patterns and processes in nature that are independent of average conditions. Jensen’s inequality is a mathematical proof that is seldom mentioned in the ecological literature but which provides a powerful tool for predicting some direct effects of environmental variance in biological systems. Qualitative predictions can be derived from the form of the relevant response functions (accelerating versus decelerating). Knowledge of the frequency distribution (especially the variance) of the driving variables allows quantitative estimates of the effects. Jensen’s inequality has relevance in every field of biology that includes nonlinear processes.

Local fractional derivative (LFD) operators have been introduced in the recent literature (Chaos 6 (1996) 505–513). Being local in nature these derivatives have proven useful in studying fractional differentiability properties of highly irregular and nowhere differentiable functions. In the present paper we prove Leibniz rule, chain rule for LFD operators. Generalization of directional LFD and multivariable fractional Taylor series to higher orders have been presented.

On a New Measure on Fractals

- A K Golmankhaneh
- D Baleanu

A. K. Golmankhaneh and D. Baleanu, "On a New Measure on Fractals", Journal of
Inequalitiesand Applications, vol. 522, no. 1, pp. 1-9, 2013.

Jensens Inequality Predicts Effects of Environmental

- J R Jonathan
- P A Mattew

J. R. Jonathan and P. A. Mattew, "Jensens Inequality Predicts Effects of Environmental

Jensen's Inequality, Parameter Uncertainty, and Multiperiod Investment

- M Grinalatt
- J T Linnainmaa

M. Grinalatt and J. T. Linnainmaa, "Jensen's Inequality, Parameter Uncertainty, and
Multiperiod Investment", Review of Asset Pricing Studies, vol. 1, no. 1, pp.1-34, 2011.