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23 pages, 504 KB

Open AccessArticle

Fractional Reverse Inequalities Involving Generic Interval-Valued Convex Functions and Applications

by Bandar Bin-Mohsin, Muhammad Zakria Javed, Muhammad Uzair Awan, Badreddine Meftah and Artion Kashuri

Fractal Fract. 2024, 8(10), 587; https://doi.org/10.3390/fractalfract8100587 - 3 Oct 2024

Cited by 9 | Viewed by 1762

Abstract

The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hölder’s inequality, unified Jensen’s inequality, and Hermite–Hadamard (

H

-

H

)-like inequalities using fractional calculus [...] Read more.

The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hölder’s inequality, unified Jensen’s inequality, and Hermite–Hadamard (

H

-

H

)-like inequalities using fractional calculus and a generic class of interval-valued convexity. We introduce the concept of

I . V

-

(⋏, ℏ)

generic class of convexity, which unifies several existing definitions of convexity. By utilizing Riemann–Liouville (R-L) fractional operators and

I . V

-

(⋏, ℏ)

convexity to derive new improvements of the

H

-

H

- and Fejer and Pachpatte-like inequalities. Our results are quite unified; by substituting the different values of parameters, we obtain a blend of new and existing inequalities. These results are fruitful for establishing bounds for

I . V

R-L integral operators. Furthermore, we discuss various implications of our findings, along with numerical examples and simulations to enhance the reliability of our results. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)

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27 pages, 474 KB

Open AccessArticle

Certain New Reverse Hölder- and Minkowski-Type Inequalities for Modified Unified Generalized Fractional Integral Operators with Extended Unified Mittag–Leffler Functions

by Wengui Yang

Fractal Fract. 2023, 7(8), 613; https://doi.org/10.3390/fractalfract7080613 - 9 Aug 2023

Cited by 6 | Viewed by 1724

Abstract

In this article, we obtain certain novel reverse Hölder- and Minkowski-type inequalities for modified unified generalized fractional integral operators (FIOs) with extended unified Mittag–Leffler functions (MLFs). The predominant results of this article generalize and extend the existing fractional Hölder- and Minkowski-type integral inequalities [...] Read more.

In this article, we obtain certain novel reverse Hölder- and Minkowski-type inequalities for modified unified generalized fractional integral operators (FIOs) with extended unified Mittag–Leffler functions (MLFs). The predominant results of this article generalize and extend the existing fractional Hölder- and Minkowski-type integral inequalities in the literature. As applications, the reverse versions of weighted Radon-, Jensen- and power mean-type inequalities for modified unified generalized FIOs with extended unified MLFs are also investigated. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)

19 pages, 301 KB

Open AccessArticle

Some New Generalizations of Reverse Hilbert-Type Inequalities via Supermultiplicative Functions

by Haytham M. Rezk, Ahmed I. Saied, Ghada AlNemer and Mohammed Zakarya

Symmetry 2022, 14(10), 2043; https://doi.org/10.3390/sym14102043 - 30 Sep 2022

Viewed by 1321

Abstract

Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain [...] Read more.

Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain rule, and Jensen’s inequality on time scales, we can establish some comprehensive and generalize a number of classical reverse Hilbert-type inequalities to a general time scale space. In time scale calculus, results are unified and extended. At the same time, the theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on symmetrical properties which play an essential role in determining the correct methods to solve inequalities. As a special case of our results when the supermultiplicative function represents the identity map, we obtain some results that have been recently published. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

17 pages, 321 KB

Open AccessArticle

Some Generalizations of the Jensen-Type Inequalities with Applications

by Mirna Rodić

Axioms 2022, 11(5), 227; https://doi.org/10.3390/axioms11050227 - 13 May 2022

Cited by 5 | Viewed by 2735

Abstract

Motivated by some results about reverses of the Jensen inequality for positive measure, in this paper we give generalizations of those results for real Stieltjes measure

d λ

which is not necessarily positive using several Green functions. Utilizing these results we define some [...] Read more.

Motivated by some results about reverses of the Jensen inequality for positive measure, in this paper we give generalizations of those results for real Stieltjes measure

d λ

which is not necessarily positive using several Green functions. Utilizing these results we define some new mean value theorems of Lagrange and Cauchy types, and derive some new Cauchy-type means. Full article

(This article belongs to the Special Issue Current Research on Mathematical Inequalities)

24 pages, 321 KB

Open AccessArticle

Some New Generalizations of Reverse Hilbert-Type Inequalities on Time Scales

by Haytham M. Rezk, Ghada AlNemer, Ahmed I. Saied, Omar Bazighifan and Mohammed Zakarya

Symmetry 2022, 14(4), 750; https://doi.org/10.3390/sym14040750 - 6 Apr 2022

Cited by 8 | Viewed by 1727

Abstract

This manuscript develops the study of reverse Hilbert-type inequalities by applying reverse Hölder inequalities on

T

. We generalize the reverse inequality of Hilbert-type with power two by replacing the power with a new power

β,

β > 1 .

The main [...] Read more.

This manuscript develops the study of reverse Hilbert-type inequalities by applying reverse Hölder inequalities on

T

. We generalize the reverse inequality of Hilbert-type with power two by replacing the power with a new power

β,

β > 1 .

The main results are proved by using Specht’s ratio, chain rule and Jensen’s inequality. Our results (when

T = N

) are essentially new. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Equations: Mathematical Models, Methods and Applications Ⅱ)

16 pages, 359 KB

Open AccessArticle

Reversing Jensen’s Inequality for Information-Theoretic Analyses

by Neri Merhav

Information 2022, 13(1), 39; https://doi.org/10.3390/info13010039 - 13 Jan 2022

Cited by 5 | Viewed by 5119

Abstract

In this work, we propose both an improvement and extensions of a reverse Jensen inequality due to Wunder et al. (2021). The new proposed inequalities are fairly tight and reasonably easy to use in a wide variety of situations, as demonstrated in several [...] Read more.

In this work, we propose both an improvement and extensions of a reverse Jensen inequality due to Wunder et al. (2021). The new proposed inequalities are fairly tight and reasonably easy to use in a wide variety of situations, as demonstrated in several application examples that are relevant to information theory. Moreover, the main ideas behind the derivations turn out to be applicable to generate bounds to expectations of multivariate convex/concave functions, as well as functions that are not necessarily convex or concave. Full article

(This article belongs to the Section Information Theory and Methodology)

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20 pages, 817 KB

Open AccessArticle

Some New Reverse Hilbert’s Inequalities on Time Scales

by Ghada AlNemer, Ahmed I. Saied, Mohammed Zakarya, Hoda A. Abd El-Hamid, Omar Bazighifan and Haytham M. Rezk

Symmetry 2021, 13(12), 2431; https://doi.org/10.3390/sym13122431 - 15 Dec 2021

Cited by 14 | Viewed by 2631

Abstract

This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for [...] Read more.

This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for reverse inequalities of Hilbert-type. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

15 pages, 296 KB

Open AccessArticle

Refined Young Inequality and Its Application to Divergences

by Shigeru Furuichi and Nicuşor Minculete

Entropy 2021, 23(5), 514; https://doi.org/10.3390/e23050514 - 23 Apr 2021

Cited by 10 | Viewed by 2846

Abstract

We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also studied some properties on the difference between the weighted arithmetic mean and the [...] Read more.

We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also studied some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the Rényi divergence, the Jeffreys–Tsallis divergence and the Jensen–Shannon–Tsallis divergence. Full article

(This article belongs to the Special Issue Types of Entropies and Divergences with Their Applications)

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