Discrete majorization type inequalities for convex functions on rectangles

Khan, Muhammad Adil; Wu, Shanhe; Ullah, Hidayat; Chu, Yu‐Ming

doi:10.1186/s13660-019-1964-3

Cited by 32 publications

(7 citation statements)

References 47 publications

(25 reference statements)

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“…There are many inequalities, which it would not be possible to establish without convex functions [28][29][30]. Some of the interesting inequalities that have been acquired by convex functions are the majorization inequality [21], Slater's inequality [31], Hermite-Hadamard's inequality [7], Jensen-Mercer's [32] inequality, and many more [33][34][35]. It has been confirmed that convex functions have played an important role in the development of the field of mathematical inequalities [36][37][38][39].…”

Section: Remarkmentioning

confidence: 99%

“…Due to the extensive applications and great importance of convex functions, this class has been generalized in various ways with the help of its characteristics and properties [16][17][18]. Some of the generalizations of convex functions are P-convex [19], s-convex [20], coordinates convex [21], h-convex [22], and 4-convex functions [23], etc. Among the above classes of convex functions, one of the significant classes is 6-convex functions.…”

Section: Introductionmentioning

confidence: 99%

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Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

et al. 2023

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The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some numerical experiments are performed to determine the significance and correctness of the intended estimates. Several outcomes of the main results are discussed for the Hölder inequality and the power and quasi-arithmetic means. Furthermore, some applications are presented in information theory, which provide some bounds for the divergences, Bhattacharyya coefficient, Shannon entropy, and Zipf–Mandelbrot entropy.

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Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

et al. 2023

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“…In 2004, Dragomir [54] also provided the weighted version of the majorization inequality by taking certain tuples with relaxed majorization conditions. In 2019, Adil Khan et al [55] presented the extension of the classical majorization inequality and its weighted versions under different circumstances for convex functions on rectangles. Wu et al [56] acquired improvements of the majorization-type inequalities for convex functions through Taylor's theorem with a mean value form of remainder.…”

Section: Introductionmentioning

confidence: 99%

Derivation of Bounds for Majorization Differences by a Novel Method and Its Applications in Information Theory

Basir,

Khan,

Ullah

et al. 2023

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In the recent era of research developments, mathematical inequalities and their applications perform a very consequential role in different aspects, and they provide an engaging area for research activities. In this paper, we propose a new approach for the improvement of the classical majorization inequality and its weighted versions in a discrete sense. The proposed improvements give several estimates for the majorization differences. Some earlier improvements of the Jensen and Slater inequalities are deduced as direct consequences of the obtained results. We also discuss the conditions under which the main results give better estimates for the majorization differences. Applications of the acquired results are also presented in information theory.

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“…Many authors have been inventing the properties and applications of strongly convex function for more information, [5][6][7][8][9][10]. In this paper we will use this definition for normed space R.…”

Section: Introductionmentioning

confidence: 99%

Refinements of two fractional versions of Hadamard inequalities for Caputo fractional derivatives and related results

Farid¹,

Rehman²,

Bibi³

et al. 2021

Open J. Math. Sci.

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The aim of this paper is to study the fractional Hadamard inequalities for Caputo fractional derivatives of strongly convex functions. We obtain refinements of two known fractional versions of the Hadamard inequality for convex functions. By applying identities for Caputo fractional derivatives we get refinements of error bounds of these inequalities. The given results simultaneously provide refinements as well as generalizations of already known inequalities.

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Discrete majorization type inequalities for convex functions on rectangles

Abstract: In this paper, we present several discrete majorization type inequalities for the convex functions defined on rectangles. MSC: 26D15; 26B25

Cited by 32 publications

References 47 publications

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

Derivation of Bounds for Majorization Differences by a Novel Method and Its Applications in Information Theory

Refinements of two fractional versions of Hadamard inequalities for Caputo fractional derivatives and related results

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