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Recent Trends in Convex Analysis and Mathematical Inequalities
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Recent Trends in Convex Analysis and Mathematical Inequalities

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 21847

Special Issue Editor

Prof. Dr. Marius Radulescu

E-Mail Website
Guest Editor
Institute of Mathematical Statistics and Applied Mathematics "Gheorghe Mihoc-Caius Iacob", Casa Academiei, Calea 13 Septembrie, nr. 13, RO-050711 Bucharest, Romania
Interests: operations research; risk management; mathematics of finance; optimization theory; nonlinear functional analysis; differential equations; real analysis; numerical analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets. It introduces analytic tools for studying convexity and provides analytical applications of the concept. Convex analysis touches almost all branches of mathematics. Convex functions play an important role in many areas of mathematics, as well as in other areas of science economy, engineering, medicine, industry, and business. It is especially important in the study of optimization problems, where it is distinguished by a number of convenient properties (for example, any minimum of a convex function is a global minimum, or the maximum is attained at a boundary point). This explains why there is a very rich theory of convex functions and convex sets. Optimization of convex functions has many practical applications (circuit design, controller design, modeling, etc.). However, any inequalities hold for convex functions. Some of them hold only for convex functions.

The famous Jensen and Hermite–Hadamard inequalities hold for convex functions. Jensen’s inequality can be used to deduce inequalities such as the arithmetic-geometric mean inequality and Hölder’s inequality. Inequalities play an important role in almost all branches of mathematics as well as in other areas of science. Today, many classical inequalities are still being improved and/or generalized by many researchers. This proves that inequalities have been and are a subject of very active research.

In this Special Issue, we call for papers on new results in the domain of convex analysis, mathematical inequalities, and applications in probability and statistics. Welcomed are new proofs of well-known inequalities, or inequalities in various domains: integral inequalities, differential inequalities, norm, operator, and matrix inequalities.

Prof. Dr. Marius Radulescu
Guest Editor

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Convex analysis
  • Convex optimization
  • Convex functions
  • Biconvex functions
  • Generalized convexity
  • Majorization theory
  • Schur convex functions
  • Jensen inequality
  • Hermite–Hadamard inequality
  • Weighted inequalities
  • Geometric inequalities
  • Variational inequalities
  • Equilibrium problems

Published Papers (21 papers)

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Research

11 pages, 250 KiB  
Article
Cauchy Problem with Summable Initial-Value Functions for Parabolic Equations with Translated Potentials
by Andrey B. Muravnik and Grigorii L. Rossovskii
Mathematics 2024, 12(6), 895; https://doi.org/10.3390/math12060895 - 18 Mar 2024
Viewed by 485
Abstract
We study the Cauchy problem for differential–difference parabolic equations with potentials undergoing translations with respect to the spatial-independent variable. Such equations are used for the modeling of various phenomena not covered by the classical theory of differential equations (such as nonlinear optics, nonclassical [...] Read more.
We study the Cauchy problem for differential–difference parabolic equations with potentials undergoing translations with respect to the spatial-independent variable. Such equations are used for the modeling of various phenomena not covered by the classical theory of differential equations (such as nonlinear optics, nonclassical diffusion, multilayer plates and envelopes, and others). From the viewpoint of the pure theory, they are important due to crucially new effects not arising in the case of differential equations and due to the fact that a number of classical methods, tools, and approaches turn out to be inapplicable in the nonlocal theory. The qualitative novelty of our investigation is that the initial-value function is assumed to be summable. Earlier, only the case of bounded (essentially bounded) initial-value functions was investigated. For the prototype problem (the spatial variable is single and the nonlocal term of the equation is single), we construct the integral representation of a solution and show its smoothness in the open half-plane. Further, we find a condition binding the coefficient at the nonlocal potential and the length of its translation such that this condition guarantees the uniform decay (weighted decay) of the constructed solution under the unbounded growth of time. The rate of this decay (weighted decay) is estimated as well. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
18 pages, 309 KiB  
Article
More General Ostrowski-Type Inequalities in the Fuzzy Context
by Muhammad Amer Latif
Mathematics 2024, 12(3), 500; https://doi.org/10.3390/math12030500 - 5 Feb 2024
Viewed by 485
Abstract
In this study, Ostrowski-type inequalities in fuzzy settings were investigated. A detailed theory of fuzzy analysis is provided and utilized to establish the Ostrowski-type inequality in the fuzzy number-valued space. The results obtained in this research not only provide a generalization of the [...] Read more.
In this study, Ostrowski-type inequalities in fuzzy settings were investigated. A detailed theory of fuzzy analysis is provided and utilized to establish the Ostrowski-type inequality in the fuzzy number-valued space. The results obtained in this research not only provide a generalization of the results of Dragomir but also give an extended version of the Ostrowski-type inequalities obtained by Anastassiou. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
16 pages, 292 KiB  
Article
Order Properties Concerning Tsallis Residual Entropy
by Răzvan-Cornel Sfetcu and Vasile Preda
Mathematics 2024, 12(3), 417; https://doi.org/10.3390/math12030417 - 27 Jan 2024
Viewed by 446
Abstract
With the help of Tsallis residual entropy, we introduce Tsallis quantile entropy order between two random variables. We give necessary and sufficient conditions, study closure and reversed closure properties under parallel and series operations and show that this order is preserved in the [...] Read more.
With the help of Tsallis residual entropy, we introduce Tsallis quantile entropy order between two random variables. We give necessary and sufficient conditions, study closure and reversed closure properties under parallel and series operations and show that this order is preserved in the proportional hazard rate model, proportional reversed hazard rate model, proportional odds model and record values model. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
21 pages, 399 KiB  
Article
Subgradient Extra-Gradient Algorithm for Pseudomonotone Equilibrium Problems and Fixed-Point Problems of Bregman Relatively Nonexpansive Mappings
by Roushanak Lotfikar, Gholamreza Zamani Eskandani, Jong-Kyu Kim and Michael Th. Rassias
Mathematics 2023, 11(23), 4821; https://doi.org/10.3390/math11234821 - 29 Nov 2023
Viewed by 617
Abstract
In this article, we introduce a new subgradient extra-gradient algorithm to find the common element of a set of fixed points of a Bregman relatively nonexpansive mapping and the solution set of an equilibrium problem involving a Pseudomonotone and Bregman–Lipschitz-type bifunction in reflexive [...] Read more.
In this article, we introduce a new subgradient extra-gradient algorithm to find the common element of a set of fixed points of a Bregman relatively nonexpansive mapping and the solution set of an equilibrium problem involving a Pseudomonotone and Bregman–Lipschitz-type bifunction in reflexive Banach spaces. The advantage of the algorithm is that it is run without prior knowledge of the Bregman–Lipschitz coefficients. Finally, two numerical experiments are reported to illustrate the efficiency of the proposed algorithm. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
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13 pages, 272 KiB  
Article
Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product
by Sorin Rădulescu, Marius Rădulescu and Mihaly Bencze
Mathematics 2023, 11(21), 4405; https://doi.org/10.3390/math11214405 - 24 Oct 2023
Viewed by 617
Abstract
The aim of this paper is to investigate when a linear normed space is an inner product space. Several conditions in a linear normed space are formulated with the help of inequalities. Some of them are from the literature and others are new. [...] Read more.
The aim of this paper is to investigate when a linear normed space is an inner product space. Several conditions in a linear normed space are formulated with the help of inequalities. Some of them are from the literature and others are new. We prove that these conditions are equivalent with the fact that the norm is induced by an inner product. One of the new results is the following: in an inner product space, the sum of opposite edges of a tetrahedron are the sides of an acute angled triangle. The converse of this result holds also. More precisely, this property characterizes inner product spaces. Another new result is the following: in a tetrahedron, the sum of squares of opposite edges are the lengths of a triangle. We prove also that this property characterizes inner product spaces. In addition, we give simpler proofs to some theorems already known from the publications of other authors. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
13 pages, 317 KiB  
Article
Prabhakar Functions of Le Roy Type: Inequalities and Asymptotic Formulae
by Jordanka Paneva-Konovska
Mathematics 2023, 11(17), 3768; https://doi.org/10.3390/math11173768 - 1 Sep 2023
Cited by 5 | Viewed by 599
Abstract
In this paper, the four-index generalization of the classical Le Roy function is considered on a wider set of parameters and its order and type are given. Letting one of the parameters take non-negative integer values, a family of functions with such a [...] Read more.
In this paper, the four-index generalization of the classical Le Roy function is considered on a wider set of parameters and its order and type are given. Letting one of the parameters take non-negative integer values, a family of functions with such a type of index is constructed. The behaviour of these functions is studied in the complex plane C and in different domains thereof. First, several inequalities are obtained in C, and then they are modified on its compact subsets as well. Moreover, an asymptotic formula is proved for ‘large’ values of the indices of these functions. Additionally, the multi-index analogue of the abovementioned four-index Le Roy type function is considered and its basic properties are obtained. Finally, several special cases of the two functions under consideration are discussed. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
28 pages, 390 KiB  
Article
Pareto Efficiency Criteria and Duality for Multiobjective Fractional Programming Problems with Equilibrium Constraints on Hadamard Manifolds
by Arnav Ghosh, Balendu Bhooshan Upadhyay and I. M. Stancu-Minasian
Mathematics 2023, 11(17), 3649; https://doi.org/10.3390/math11173649 - 23 Aug 2023
Cited by 2 | Viewed by 636
Abstract
This article deals with multiobjective fractional programming problems with equilibrium constraints in the setting of Hadamard manifolds (abbreviated as MFPPEC). The generalized Guignard constraint qualification (abbreviated as GGCQ) for MFPPEC is presented. Furthermore, the Karush–Kuhn–Tucker (abbreviated as KKT) type necessary criteria of Pareto [...] Read more.
This article deals with multiobjective fractional programming problems with equilibrium constraints in the setting of Hadamard manifolds (abbreviated as MFPPEC). The generalized Guignard constraint qualification (abbreviated as GGCQ) for MFPPEC is presented. Furthermore, the Karush–Kuhn–Tucker (abbreviated as KKT) type necessary criteria of Pareto efficiency for MFPPEC are derived using GGCQ. Sufficient criteria of Pareto efficiency for MFPPEC are deduced under some geodesic convexity hypotheses. Subsequently, Mond–Weir and Wolfe type dual models related to MFPPEC are formulated. The weak, strong, and strict converse duality results are derived relating MFPPEC and the respective dual models. Suitable nontrivial examples have been furnished to demonstrate the significance of the results established in this article. The results derived in the article extend and generalize several notable results previously existing in the literature. To the best of our knowledge, optimality conditions and duality for MFPPEC have not yet been studied in the framework of manifolds. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
13 pages, 286 KiB  
Article
A Weighted Generalization of Hardy–Hilbert-Type Inequality Involving Two Partial Sums
by Bicheng Yang and Shanhe Wu
Mathematics 2023, 11(14), 3212; https://doi.org/10.3390/math11143212 - 21 Jul 2023
Cited by 1 | Viewed by 517
Abstract
In this paper, we address Hardy–Hilbert-type inequality by virtue of constructing weight coefficients and introducing parameters. By using the Euler–Maclaurin summation formula, Abel’s partial summation formula, and differential mean value theorem, a new weighted Hardy–Hilbert-type inequality containing two partial sums can be proven, [...] Read more.
In this paper, we address Hardy–Hilbert-type inequality by virtue of constructing weight coefficients and introducing parameters. By using the Euler–Maclaurin summation formula, Abel’s partial summation formula, and differential mean value theorem, a new weighted Hardy–Hilbert-type inequality containing two partial sums can be proven, which is a further generalization of an existing result. Based on the obtained results, we provide the equivalent statements of the best possible constant factor related to several parameters. Also, we illustrate how the inequalities obtained in the main results can generate some new Hardy–Hilbert-type inequalities. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
14 pages, 309 KiB  
Article
On Sub Convexlike Optimization Problems
by Renying Zeng
Mathematics 2023, 11(13), 2928; https://doi.org/10.3390/math11132928 - 29 Jun 2023
Viewed by 608
Abstract
In this paper, we show that the sub convexlikeness and subconvexlikeness defined by V. Jeyakumar are equivalent in locally convex topological spaces. We also deal with set-valued vector optimization problems and obtained vector saddle-point theorems and vector Lagrangian theorems. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
16 pages, 363 KiB  
Article
Bounds for the Error in Approximating a Fractional Integral by Simpson’s Rule
by Hüseyin Budak, Fatih Hezenci, Hasan Kara and Mehmet Zeki Sarikaya
Mathematics 2023, 11(10), 2282; https://doi.org/10.3390/math11102282 - 13 May 2023
Cited by 2 | Viewed by 897
Abstract
Simpson’s rule is a numerical method used for approximating the definite integral of a function. In this paper, by utilizing mappings whose second derivatives are bounded, we acquire the upper and lower bounds for the Simpson-type inequalities by means of Riemann–Liouville fractional integral [...] Read more.
Simpson’s rule is a numerical method used for approximating the definite integral of a function. In this paper, by utilizing mappings whose second derivatives are bounded, we acquire the upper and lower bounds for the Simpson-type inequalities by means of Riemann–Liouville fractional integral operators. We also study special cases of our main results. Furthermore, we give some examples with graphs to illustrate the main results. This study on fractional Simpson’s inequalities is the first paper in the literature as a method. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
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23 pages, 541 KiB  
Article
Properties of Coordinated h1,h2-Convex Functions of Two Variables Related to the Hermite–Hadamard–Fejér Type Inequalities
by Muhammad Amer Latif
Mathematics 2023, 11(5), 1201; https://doi.org/10.3390/math11051201 - 28 Feb 2023
Cited by 1 | Viewed by 1062
Abstract
In this paper, we prove the Hermite–Hadamard–Fejér type inequalities for coordinated h1,h2-convex functions on the rectangle from the plane R2. Some generalizations of the Hermite–Hadamard-type inequalities of two variables are also obtained as a consequence. Some [...] Read more.
In this paper, we prove the Hermite–Hadamard–Fejér type inequalities for coordinated h1,h2-convex functions on the rectangle from the plane R2. Some generalizations of the Hermite–Hadamard-type inequalities of two variables are also obtained as a consequence. Some properties of two functionals which are connected with the coordinated h1,h2-convex functions are provided as well. Finally, we give applications of the acquired results to special means of positive real numbers. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
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11 pages, 341 KiB  
Article
Optimal Control Problem for Minimization of Net Energy Consumption at Metro
by Constantin Udriste, Ionel Tevy and Paun Antonescu
Mathematics 2023, 11(4), 1035; https://doi.org/10.3390/math11041035 - 18 Feb 2023
Viewed by 996
Abstract
The optimal control currently decides the minimum energy consumption within the problems attached to subways. Among other things, we formulate and solve an optimal bi-control problem, the two controls being the acceleration and the feed-back of a Riemannian connection. The control space is [...] Read more.
The optimal control currently decides the minimum energy consumption within the problems attached to subways. Among other things, we formulate and solve an optimal bi-control problem, the two controls being the acceleration and the feed-back of a Riemannian connection. The control space is a square, and the optimal controls are of the bang–bang type. The third component of the optimal solution is the maximum value function, as a solution of the Hamilton–Jacobi–Bellman PDE. The examples of energy optimal trajectories refer to the lines of the Bucharest subway. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
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19 pages, 333 KiB  
Article
Some Companions of Fejér Type Inequalities Using GA-Convex Functions
by Muhammad Amer Latif
Mathematics 2023, 11(2), 392; https://doi.org/10.3390/math11020392 - 11 Jan 2023
Cited by 5 | Viewed by 1043
Abstract
In this paper, we present some new and novel mappings defined over 0,1 with the help of GA-convex functions. As a consequence, we obtain companions of Fejér-type inequalities for GA-convex functions with the help of these mappings, [...] Read more.
In this paper, we present some new and novel mappings defined over 0,1 with the help of GA-convex functions. As a consequence, we obtain companions of Fejér-type inequalities for GA-convex functions with the help of these mappings, which provide refinements of some known results. The properties of these mappings are discussed as well. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
21 pages, 350 KiB  
Article
Jensen–Mercer and Hermite–Hadamard–Mercer Type Inequalities for GA-h-Convex Functions and Its Subclasses with Applications
by Asfand Fahad, Ayesha, Yuanheng Wang and Saad Ihsaan Butt
Mathematics 2023, 11(2), 278; https://doi.org/10.3390/math11020278 - 5 Jan 2023
Cited by 4 | Viewed by 1177
Abstract
Many researchers have been attracted to the study of convex analysis theory due to both facts, theoretical significance, and the applications in optimization, economics, and other fields, which has led to numerous improvements and extensions of the subject over the years. An essential [...] Read more.
Many researchers have been attracted to the study of convex analysis theory due to both facts, theoretical significance, and the applications in optimization, economics, and other fields, which has led to numerous improvements and extensions of the subject over the years. An essential part of the theory of mathematical inequalities is the convex function and its extensions. In the recent past, the study of Jensen–Mercer inequality and Hermite–Hadamard–Mercer type inequalities has remained a topic of interest in mathematical inequalities. In this paper, we study several inequalities for GA-h-convex functions and its subclasses, including GA-convex functions, GA-s-convex functions, GA-Q-convex functions, and GA-P-convex functions. We prove the Jensen–Mercer inequality for GA-h-convex functions and give weighted Hermite–Hadamard inequalities by applying the newly established Jensen–Mercer inequality. We also establish inequalities of Hermite–Hadamard–Mercer type. Thus, we give new insights and variants of Jensen–Mercer and related inequalities for GA-h-convex functions. Furthermore, we apply our main results along with Hadamard fractional integrals to prove weighted Hermite–Hadamard–Mercer inequalities for GA-h-convex functions and its subclasses. As special cases of the proven results, we capture several well-known results from the relevant literature. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
17 pages, 366 KiB  
Article
Some Novel Estimates of Hermite–Hadamard and Jensen Type Inequalities for (h1,h2)-Convex Functions Pertaining to Total Order Relation
by Tareq Saeed, Waqar Afzal, Khurram Shabbir, Savin Treanţă and Manuel De la Sen
Mathematics 2022, 10(24), 4777; https://doi.org/10.3390/math10244777 - 15 Dec 2022
Cited by 11 | Viewed by 1568
Abstract
There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order [...] Read more.
There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order relations are quite different and novel in the literature and are calculated as ω=⟨ωc,ωr⟩=⟨ω¯+ω̲2,ω¯−ω̲2⟩. A major benefit of total order relations is that they produce more efficient results than other order relations. This study introduces the notion of CR-(h1,h2)-convex function using total order relations. Center and Radius order relations are a powerful tool for studying inequalities based on their properties and widespread application. Using this novel notion, we first developed some variants of Hermite–Hadamard inequality and then constructed Jensen inequality. Based on the results, this new concept is extremely useful in connection with a variety of inequalities. There are many new and well-known convex functions unified by this type of convexity. These results will stimulate further research on inequalities for fractional interval-valued functions and fuzzy interval-valued functions, as well as the optimization problems associated with them. For the purpose of verifying our main findings, we provide some nontrivial examples. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
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21 pages, 350 KiB  
Article
Some New Mathematical Integral Inequalities Pertaining to Generalized Harmonic Convexity with Applications
by Muhammad Tariq, Soubhagya Kumar Sahoo, Sotiris K. Ntouyas, Omar Mutab Alsalami, Asif Ali Shaikh and Kamsing Nonlaopon
Mathematics 2022, 10(18), 3286; https://doi.org/10.3390/math10183286 - 10 Sep 2022
Viewed by 888
Abstract
The subject of convex analysis and integral inequalities represents a comprehensive and absorbing field of research within the field of mathematical interpretation. In recent times, the strategies of convex theory and integral inequalities have become the subject of intensive research at historical and [...] Read more.
The subject of convex analysis and integral inequalities represents a comprehensive and absorbing field of research within the field of mathematical interpretation. In recent times, the strategies of convex theory and integral inequalities have become the subject of intensive research at historical and contemporary times because of their applications in various branches of sciences. In this work, we reveal the idea of a new version of generalized harmonic convexity i.e., an m–polynomial p–harmonic s–type convex function. We discuss this new idea by employing some examples and demonstrating some interesting algebraic properties. Furthermore, this work leads us to establish some new generalized Hermite–Hadamard- and generalized Ostrowski-type integral identities. Additionally, employing Hölder’s inequality and the power-mean inequality, we present some refinements of the H–H (Hermite–Hadamard) inequality and Ostrowski inequalities. Finally, we investigate some applications to special means involving the established results. These new results yield us some generalizations of the prior results in the literature. We believe that the methodology and concept examined in this paper will further inspire interested researchers. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
24 pages, 379 KiB  
Article
On the Probability of Finding Extremes in a Random Set
by Anișoara Maria Răducan, Constanța Zoie Rădulescu, Marius Rădulescu and Gheorghiță Zbăganu
Mathematics 2022, 10(10), 1623; https://doi.org/10.3390/math10101623 - 10 May 2022
Cited by 2 | Viewed by 1118
Abstract
We consider a sequence Zjj≥1 of i.i.d. d-dimensional random vectors and for every n≥1 consider the sample Sn={Z1,Z2,…,Zn}. We say that [...] Read more.
We consider a sequence Zjj≥1 of i.i.d. d-dimensional random vectors and for every n≥1 consider the sample Sn={Z1,Z2,…,Zn}. We say that Zj is a “leader” in the sample Sn if Zj≥Zk,∀k∈{1,2,…,n}and Zj is an “anti-leader” if Zj≤Zk,∀k∈{1,2,…,n}. After all, the leader and the anti-leader are the naive extremes. Let an be the probability that Sn has a leader, bn be the probability that Sn has an anti-leader and cn be the probability that Sn has both a leader and an anti-leader. One of the aims of the paper is to compute, or, at least to estimate, or if even that is not possible, to estimate the limits of this quantities. Another goal is to find conditions on the distribution F of Zjj≥1 so that the inferior limits of an,bn,cn are positive. We give examples of distributions for which we can compute these probabilities and also examples when we are not able to do that. Then we establish conditions, unfortunately only sufficient when the limits are positive. Doing that we discovered a lot of open questions and we make two annoying conjectures—annoying because they seemed to be obvious but at a second thought we were not able to prove them. It seems that these problems have never been approached in the literature. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
19 pages, 336 KiB  
Article
Improvements of Slater’s Inequality by Means of 4-Convexity and Its Applications
by Xuexiao You, Muhammad Adil Khan, Hidayat Ullah and Tareq Saeed
Mathematics 2022, 10(8), 1274; https://doi.org/10.3390/math10081274 - 12 Apr 2022
Cited by 8 | Viewed by 1317
Abstract
In 2021, Ullah et al., introduced a new approach for the derivation of results for Jensen’s inequality. The purpose of this article, is to use the same technique and to derive improvements of Slater’s inequality. The planned improvements are demonstrated in both discrete [...] Read more.
In 2021, Ullah et al., introduced a new approach for the derivation of results for Jensen’s inequality. The purpose of this article, is to use the same technique and to derive improvements of Slater’s inequality. The planned improvements are demonstrated in both discrete as well as in integral versions. The quoted results allow us to provide relationships for the power means. Moreover, with the help of established results, we present some estimates for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and Bhattacharyya coefficient. In addition, we discuss some additional applications of the main results for the Zipf–Mandelbrot entropy. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
12 pages, 259 KiB  
Article
Jensen Functional, Quasi-Arithmetic Mean and Sharp Converses of Hölder’s Inequalities
by Slavko Simić and Vesna Todorčević
Mathematics 2021, 9(23), 3104; https://doi.org/10.3390/math9233104 - 1 Dec 2021
Cited by 6 | Viewed by 1533
Abstract
In this article, we give sharp two-sided bounds for the generalized Jensen functional Jn(f,g,h;p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean [...] Read more.
In this article, we give sharp two-sided bounds for the generalized Jensen functional Jn(f,g,h;p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean An(h;p,x). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
11 pages, 269 KiB  
Article
Hermite–Hadamard–Mercer-Type Inequalities for Harmonically Convex Mappings
by Xuexiao You, Muhammad Aamir Ali, Hüseyin Budak, Jiraporn Reunsumrit and Thanin Sitthiwirattham
Mathematics 2021, 9(20), 2556; https://doi.org/10.3390/math9202556 - 12 Oct 2021
Cited by 9 | Viewed by 1299
Abstract
In this paper, we prove Hermite–Hadamard–Mercer inequalities, which is a new version of the Hermite–Hadamard inequalities for harmonically convex functions. We also prove Hermite–Hadamard–Mercer-type inequalities for functions whose first derivatives in absolute value are harmonically convex. Finally, we discuss how special means can [...] Read more.
In this paper, we prove Hermite–Hadamard–Mercer inequalities, which is a new version of the Hermite–Hadamard inequalities for harmonically convex functions. We also prove Hermite–Hadamard–Mercer-type inequalities for functions whose first derivatives in absolute value are harmonically convex. Finally, we discuss how special means can be used to address newly discovered inequalities. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
18 pages, 318 KiB  
Article
Some New Simpson’s and Newton’s Formulas Type Inequalities for Convex Functions in Quantum Calculus
by Pimchana Siricharuanun, Samet Erden, Muhammad Aamir Ali, Hüseyin Budak, Saowaluck Chasreechai and Thanin Sitthiwirattham
Mathematics 2021, 9(16), 1992; https://doi.org/10.3390/math9161992 - 20 Aug 2021
Cited by 10 | Viewed by 1888
Abstract
In this paper, using the notions of qκ2-quantum integral and qκ2-quantum derivative, we present some new identities that enable us to obtain new quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions. This paper, [...] Read more.
In this paper, using the notions of qκ2-quantum integral and qκ2-quantum derivative, we present some new identities that enable us to obtain new quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions. This paper, in particular, generalizes and expands previous findings in the field of quantum and classical integral inequalities obtained by various authors. Full article
(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)
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