Advances in Mathematical Inequalities and Applications

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 May 2022) | Viewed by 20316

Special Issue Editor


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Guest Editor
Department of Mathematics, University of Split, 21000 Split, Croatia
Interests: inequalities; real analysis; mathematical analysis

Special Issue Information

Dear Colleagues,

Why do we study inequalities? The answer to this question was given by Bellman in a very concrete and elegant fashion: "There are three reasons for the study of inequalities: practical, theoretical and aesthetic. In many practical investigations, it is necessary to bound one quantity by another. The classical inequalities are very useful for this purpose. From the theoretical point of view, very simple questions give rise to entire theories. For example, we may ask when the nonnegativity of one quantity implies that to another. This simple question leads to the theory of positive operators and theory of differential inequalities. Another question which gives rise to much interesting research is that of finding equalities associated with inequalities. We use the principle that every inequality should come from an equality which makes the inequality obvious. Along these lines, we may also look for representation which make inequalities obvious. Often, these representations are maxima or minima of certain quantities. Finally, let us turn to aesthetic aspects. As has been pointed out, beauty is in the eyes of the beholder. However, it is generally agreed that certain pieces of music, art or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive."
In this Special Issue, we present new results related to some classical inequalities such as the Jensen inequality, Jensen–Steffensen inequality, Jessen inequality, Grüss inequality, Chebyshev inequality, etc. They have various applications in other branches of mathematics, such as numerical analysis, probability and statistics, as well as in other sciences such as information theory.

Prof. Dr. Milica Klaricic Bakula
Guest Editor

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Keywords

  • inequalities
  • convex functions
  • generalized convexity
  • Jensen inequality
  • Jensen–Steffensen inequality
  • Jessen inequality
  • Hermite–Hadamard inequalities
  • Grüss inequality
  • Chebyshev inequality
  • means
  • entropy
  • Zipf–Mandelbrot law
  • quadrature formulae

Published Papers (11 papers)

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Editorial

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4 pages, 184 KiB  
Editorial
Advances in Mathematical Inequalities and Applications
by Milica Klaričić Bakula
Mathematics 2023, 11(10), 2397; https://doi.org/10.3390/math11102397 - 22 May 2023
Viewed by 1303
Abstract
Why do we study inequalities [...] Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)

Research

Jump to: Editorial

9 pages, 278 KiB  
Article
The Development of Suitable Inequalities and Their Application to Systems of Logical Equations
by Dostonjon Numonjonovich Barotov, Ruziboy Numonjonovich Barotov, Vladimir Soloviev, Vadim Feklin, Dilshod Muzafarov, Trusunboy Ergashboev and Khudoyberdi Egamov
Mathematics 2022, 10(11), 1851; https://doi.org/10.3390/math10111851 - 27 May 2022
Cited by 5 | Viewed by 1552
Abstract
In this paper, two not-difficult inequalities are invented and proved in detail, which adequately describe the behavior of discrete logical functions xor(x1, x2,, xn) and [...] Read more.
In this paper, two not-difficult inequalities are invented and proved in detail, which adequately describe the behavior of discrete logical functions xor(x1, x2,, xn) and and(x1, x2,, xn). Based on these proven inequalities, infinitely differentiable extensions of the logical functions xor(x1, x2,, xn) and and(x1, x2,, xn) were defined for the entire n. These suitable extensions were applied to systems of logical equations. Specifically, the system of m logical equations in a constructive way without adding any equations (not field equations and no others) is transformed in n first into an equivalent system of m smooth rational equations (SmSRE) so that the solution of SmSRE can be reduced to the problem minimization of the objective function, and any numerical optimization methods can be applied since the objective function will be infinitely differentiable. Again, we transformed SmSRE into an equivalent system of m polynomial equations (SmPE). This means that any symbolic methods for solving polynomial systems can be used to solve and analyze an equivalent SmPE. The equivalence of these systems has been proved in detail. Based on these proofs and results, in the next paper, we plan to study the practical applicability of numerical optimization methods for SmSRE and symbolic methods for SmPE. Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)
18 pages, 292 KiB  
Article
Hermite–Hadamard-Type Inequalities and Two-Point Quadrature Formula
by Josipa Barić
Mathematics 2022, 10(9), 1432; https://doi.org/10.3390/math10091432 - 24 Apr 2022
Viewed by 1205
Abstract
As convexity plays an important role in many aspects of mathematical programming, e.g., for obtaining sufficient optimality conditions and in duality theorems, and one of the most important inequalities for convex functions is the Hermite–Hadamard inequality, the importance of this paper lies in [...] Read more.
As convexity plays an important role in many aspects of mathematical programming, e.g., for obtaining sufficient optimality conditions and in duality theorems, and one of the most important inequalities for convex functions is the Hermite–Hadamard inequality, the importance of this paper lies in providing some new improvements for convex functions and new directions in studying new variants of the Hermite–Hadamard inequality. The first part of the article includes some known concepts regarding convex functions and related inequalities. In the second part of the study, a derivation of the Hermite–Hadamard inequality for convex functions of higher order is given, emphasizing the purpose and importance of some quadrature formulas. In the third section, the applications of the main results are presented by obtaining Hermite–Hadamard-type estimates for various classical quadrature formulas such as the Gauss–Legendre two-point quadrature formula and the Gauss–Chebyshev two-point quadrature formulas of the first and second kind. Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)
17 pages, 304 KiB  
Article
On Ostrowski Type Inequalities via the Extended Version of Montgomery’s Identity
by Asif R. Khan, Hira Nabi and Josip E. Pečarić
Mathematics 2022, 10(7), 1113; https://doi.org/10.3390/math10071113 - 30 Mar 2022
Cited by 1 | Viewed by 1276
Abstract
In this paper, we obtain new Ostrowski type inequalities by using the extended version of Montgomery identity and Green’s functions. We also give estimations of the difference between two integral means. Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)
14 pages, 296 KiB  
Article
On the Converse Jensen-Type Inequality for Generalized f-Divergences and Zipf–Mandelbrot Law
by Mirna Rodić
Mathematics 2022, 10(6), 947; https://doi.org/10.3390/math10060947 - 16 Mar 2022
Cited by 2 | Viewed by 1459
Abstract
Motivated by some recent investigations about the sharpness of the Jensen inequality, this paper deals with the sharpness of the converse of the Jensen inequality. These results are then used for deriving new inequalities for different types of generalized f-divergences. As divergences [...] Read more.
Motivated by some recent investigations about the sharpness of the Jensen inequality, this paper deals with the sharpness of the converse of the Jensen inequality. These results are then used for deriving new inequalities for different types of generalized f-divergences. As divergences measure the differences between probability distributions, these new inequalities are then applied on the Zipf–Mandelbrot law as a special kind of a probability distribution. Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)
9 pages, 260 KiB  
Article
An Application of Hayashi’s Inequality for Differentiable Functions
by Mohammad W. Alomari and Milica Klaričić Bakula
Mathematics 2022, 10(6), 907; https://doi.org/10.3390/math10060907 - 11 Mar 2022
Cited by 1 | Viewed by 1439
Abstract
In this work, we offer new applications of Hayashi’s inequality for differentiable functions by proving new error estimates of the Ostrowski- and trapezoid-type quadrature rules. Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)
9 pages, 333 KiB  
Article
A Note of Jessen’s Inequality and Their Applications to Mean-Operators
by Gul I Hina Aslam, Amjad Ali and Khaled Mehrez
Mathematics 2022, 10(6), 879; https://doi.org/10.3390/math10060879 - 10 Mar 2022
Viewed by 1622
Abstract
A variant of Jessen’s type inequality for a semigroup of positive linear operators, defined on a Banach lattice algebra, is obtained. The corresponding mean value theorems lead to a new family of mean-operators. Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)
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11 pages, 262 KiB  
Article
Refinements of the Converse Hölder and Minkowski Inequalities
by Josip Pečarić, Jurica Perić and Sanja Varošanec
Mathematics 2022, 10(2), 202; https://doi.org/10.3390/math10020202 - 10 Jan 2022
Viewed by 2021
Abstract
We give a refinement of the converse Hölder inequality for functionals using an interpolation result for Jensen’s inequality. Additionally, we obtain similar improvements of the converse of the Beckenbach inequality. We consider the converse Minkowski inequality for functionals and of its continuous form [...] Read more.
We give a refinement of the converse Hölder inequality for functionals using an interpolation result for Jensen’s inequality. Additionally, we obtain similar improvements of the converse of the Beckenbach inequality. We consider the converse Minkowski inequality for functionals and of its continuous form and give refinements of it. Application on integral mixed means is given. Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)
16 pages, 303 KiB  
Article
Chebyshev-Steffensen Inequality Involving the Inner Product
by Milica Klaričić Bakula and Josip Pečarić
Mathematics 2022, 10(1), 122; https://doi.org/10.3390/math10010122 - 1 Jan 2022
Viewed by 1258
Abstract
In this paper, we prove the Chebyshev-Steffensen inequality involving the inner product on the real m-space. Some upper bounds for the weighted Chebyshev-Steffensen functional, as well as the Jensen-Steffensen functional involving the inner product under various conditions, are also given. Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)
8 pages, 262 KiB  
Article
Jensen-Type Inequalities for (h, g; m)-Convex Functions
by Maja Andrić
Mathematics 2021, 9(24), 3312; https://doi.org/10.3390/math9243312 - 19 Dec 2021
Cited by 4 | Viewed by 2350
Abstract
Jensen-type inequalities for the recently introduced new class of (h,g;m)-convex functions are obtained, and certain special results are indicated. These results generalize and extend corresponding inequalities for the classes of convex functions that already exist in [...] Read more.
Jensen-type inequalities for the recently introduced new class of (h,g;m)-convex functions are obtained, and certain special results are indicated. These results generalize and extend corresponding inequalities for the classes of convex functions that already exist in the literature. Schur-type inequalities are given. Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)
29 pages, 396 KiB  
Article
Determination of Bounds for the Jensen Gap and Its Applications
by Hidayat Ullah, Muhammad Adil Khan and Tareq Saeed
Mathematics 2021, 9(23), 3132; https://doi.org/10.3390/math9233132 - 5 Dec 2021
Cited by 17 | Viewed by 3075
Abstract
The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature [...] Read more.
The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature on mathematical inequalities. The main intention of this article is to find some novel bounds for the Jensen difference while using some classes of twice differentiable convex functions. We obtain the proposed bounds by utilizing the power mean and Höilder inequalities, the notion of convexity and the prominent Jensen inequality for concave function. We deduce several inequalities for power and quasi-arithmetic means as a consequence of main results. Furthermore, we also establish different improvements for Hölder inequality with the help of obtained results. Moreover, we present some applications of the main results in information theory. Full article
(This article belongs to the Special Issue Advances in Mathematical Inequalities and Applications)
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