A Local Fractional Integral Inequality on Fractal Space Analogous to Anderson’s Inequality

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.

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Some k-fractional extension of Grüss-type inequalities via generalized Hilfer–Katugampola derivative

Advances in Difference Equations ◽

10.1186/s13662-020-03187-7 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Samaira Naz ◽

Muhammad Nawaz Naeem ◽

Yu-Ming Chu

Keyword(s):

Integral Inequality ◽

Fractional Integral ◽

Integral Inequalities ◽

Derivative Operator ◽

Practical Applications ◽

Mathematical Problems ◽

Grüss Type Inequalities

AbstractIn this paper, we prove several inequalities of the Grüss type involving generalized k-fractional Hilfer–Katugampola derivative. In 1935, Grüss demonstrated a fascinating integral inequality, which gives approximation for the product of two functions. For these functions, we develop some new fractional integral inequalities. Our results with this new derivative operator are capable of evaluating several mathematical problems relevant to practical applications.

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Hadamard type local fractional integral inequalities for generalized harmonically convex functions and applications

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.6319 ◽

2020 ◽

Vol 43 (9) ◽

pp. 5776-5787

Author(s):

Wenbing Sun ◽

Qiong Liu

Keyword(s):

Convex Functions ◽

Fractional Integral ◽

Integral Inequalities ◽

Local Fractional Integral ◽

Harmonically Convex Functions

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Applications of Yang-Fourier Transform to Local Fractional Equations with Local Fractional Derivative and Local Fractional Integral

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.461.306 ◽

2012 ◽

Vol 461 ◽

pp. 306-310 ◽

Cited By ~ 18

Author(s):

Wei Ping Zhong ◽

Feng Gao ◽

Xiao Ming Shen

Keyword(s):

Fourier Transform ◽

Fractional Derivative ◽

Fractional Integral ◽

Fractional Fourier Transform ◽

Local Fractional Derivative ◽

Fractional Equations ◽

Local Fractional Integral ◽

Fractal Space

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

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Generalizations of Hölder’s and Some Related Integral Inequalities on Fractal Space

Journal of Function Spaces and Applications ◽

10.1155/2013/198405 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 10

Author(s):

Guang-Sheng Chen

Keyword(s):

Fractional Calculus ◽

Integral Inequality ◽

Hölder’S Inequality ◽

Integral Inequalities ◽

Hölder's Inequality ◽

Local Fractional Calculus ◽

Related Integral ◽

Fractal Space

Based on the local fractional calculus, we establish some new generalizations of Hölder’s inequality. By using it, some related results on the generalized integral inequality in fractal space are investigated in detail.

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General Raina fractional integral inequalities on coordinates of convex functions

Advances in Difference Equations ◽

10.1186/s13662-021-03241-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Dumitru Baleanu ◽

Artion Kashuri ◽

Pshtiwan Othman Mohammed ◽

Badreddine Meftah

Keyword(s):

Mathematical Model ◽

Fractional Calculus ◽

Convex Function ◽

Mathematical Analysis ◽

Integral Inequality ◽

Convex Functions ◽

Fractional Integral ◽

Integral Inequalities ◽

Differentiable Functions ◽

Special Cases

AbstractIntegral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In this study, authors have established some generalized Raina fractional integral inequalities using an $(l_{1},h_{1})$ ( l 1 , h 1 ) -$(l_{2},h_{2})$ ( l 2 , h 2 ) -convex function on coordinates. Also, we obtain an integral identity for partial differentiable functions. As an effect of this result, two interesting integral inequalities for the $(l_{1},h_{1})$ ( l 1 , h 1 ) -$(l_{2},h_{2})$ ( l 2 , h 2 ) -convex function on coordinates are given. Finally, we can say that our findings recapture some recent results as special cases.

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Necessary and sufficient conditions on weighted multilinear fractional integral inequality

Communications on Pure and Applied Analysis ◽

10.3934/cpaa.2021196 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Zhou Yongliang ◽

Deng Yangkendi ◽

Wu Di ◽

Yan Dunyan

Keyword(s):

Integral Inequality ◽

Sobolev Inequality ◽

Fractional Integral ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Integral Inequalities ◽

Multilinear Fractional Integral ◽

Sufficient And Necessary Conditions ◽

Necessary And Sufficient

<p style='text-indent:20px;'>We consider certain kinds of weighted multi-linear fractional integral inequalities which can be regarded as extensions of the Hardy-Littlewood-Sobolev inequality. For a particular case, we characterize the sufficient and necessary conditions which ensure that the corresponding inequality holds. For the general case, we give some sufficient conditions and necessary conditions, respectively.</p>

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

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02735-3 ◽

2022 ◽

Vol 2022 (1) ◽

Author(s):

Saad Ihsan Butt ◽

Praveen Agarwal ◽

Saba Yousaf ◽

Juan L. G. Guirao

Keyword(s):

Probability Density Function ◽

Convex Function ◽

Fractional Integral ◽

Type Inequality ◽

Fractional Integrals ◽

Fractal Sets ◽

Local Fractional Integral ◽

Fractal Space ◽

Harmonic Convex ◽

Harmonically Convex Function

AbstractIn 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.

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