Mathematics

Research

8 pages, 228 KiB

Open AccessArticle

A Unified Version of Weighted Weak-Type Inequalities for the One-Sided Hardy–Littlewood Maximal Function in Orlicz Classes

by Erxin Zhang

Mathematics 2024, 12(18), 2814; https://doi.org/10.3390/math12182814 - 11 Sep 2024

Viewed by 391

Abstract

Let

M_{g}^{+} f

be the one-sided Hardy–Littlewood maximal function,

φ_{1}

be a nonnegative and nondecreasing function on

[0, \infty)

,

γ

be a positive and nondecreasing function defined on

[0, \infty)

; let [...] Read more.

Let

M_{g}^{+} f

be the one-sided Hardy–Littlewood maximal function,

φ_{1}

be a nonnegative and nondecreasing function on

[0, \infty)

,

γ

be a positive and nondecreasing function defined on

[0, \infty)

; let

φ_{2}

be a quasi-convex function and

u, v, w

be three weight functions. In this paper, we present necessary and sufficient conditions on weight functions

(u, v, w)

such that the inequality

φ_{1} (λ) \int_{{M_{g}^{+} f > λ}} u (x) g (x) d x \leq C \int_{- \infty}^{+ \infty} φ_{2} (C \frac{| f (x) | v (x)}{γ (λ)}) w (x) g (x) d x

holds. Then, we unify the weak and extra-weak-type one-sided Hardy–Littlewood maximal inequalities in the above inequality. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

12 pages, 263 KiB

Open AccessArticle

On a Discrete Version of the Hardy–Littlewood–Polya Inequality Involving Multiple Parameters in the Whole Plane

by Bicheng Yang and Shanhe Wu

Mathematics 2024, 12(15), 2319; https://doi.org/10.3390/math12152319 - 24 Jul 2024

Viewed by 537

Abstract

In this paper, by introducing multiple parameters, we establish a discrete version of the Hardy–Littlewood–Polya inequality in the whole plane. For the obtained inequality, we give the equivalent statements of the best possible constant factor linked to the parameters and deal with the [...] Read more.

In this paper, by introducing multiple parameters, we establish a discrete version of the Hardy–Littlewood–Polya inequality in the whole plane. For the obtained inequality, we give the equivalent statements of the best possible constant factor linked to the parameters and deal with the equivalent inequalities. Our main result provided a new generalization of Hardy–Littlewood–Polya inequality, and as a consequence, we show that some new inequalities of the Hardy–Littlewood–Polya type can be derived from the current results by taking the special values of parameters. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

11 pages, 235 KiB

Open AccessArticle

Criteria of a Two-Weight, Weak-Type Inequality in Orlicz Classes for Maximal Functions Defined on Homogeneous Spaces

by Erxin Zhang

Mathematics 2024, 12(14), 2271; https://doi.org/10.3390/math12142271 - 20 Jul 2024

Cited by 1 | Viewed by 461

Abstract

In this study, some new necessary and sufficient conditions for a two-weight, weak-type maximal inequality of the form [...] Read more.

In this study, some new necessary and sufficient conditions for a two-weight, weak-type maximal inequality of the form

φ_{1} (λ) \int_{{x \in X : M f (x) > λ}} ϱ (x) d μ (x) \leq c \int_{X} φ_{2} (c | f (x) |) σ (x) d μ (x)

are obtained in Orlicz classes, where

M f

is a Hardy–Littlewood maximal function defined on homogeneous spaces and

ϱ

is a weight function. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

29 pages, 514 KiB

Open AccessArticle

Symmetric Quantum Inequalities on Finite Rectangular Plane

by Saad Ihsan Butt, Muhammad Nasim Aftab and Youngsoo Seol

Mathematics 2024, 12(10), 1517; https://doi.org/10.3390/math12101517 - 13 May 2024

Cited by 1 | Viewed by 888

Abstract

Finding the range of coordinated convex functions is yet another application for the symmetric Hermite–Hadamard inequality. For any two-dimensional interval

[a_{0}, a_{1}] \times [c_{0}, c_{1}] \subseteq ℜ^{2}

, we introduce the notion [...] Read more.

Finding the range of coordinated convex functions is yet another application for the symmetric Hermite–Hadamard inequality. For any two-dimensional interval

[a_{0}, a_{1}] \times [c_{0}, c_{1}] \subseteq ℜ^{2}

, we introduce the notion of partial

q_{θ}

-,

q_{ϕ}

-, and

q_{θ} q_{ϕ}

-symmetric derivatives and a

q_{θ} q_{ϕ}

-symmetric integral. Moreover, we will construct the

q_{θ} q_{ϕ}

-symmetric Hölder’s inequality, the symmetric quantum Hermite–Hadamard inequality for the function of two variables in a rectangular plane, and address some of its related applications. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

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11 pages, 250 KiB

Open AccessArticle

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 788

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

Open AccessArticle

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

Cited by 1 | Viewed by 822

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

Open AccessArticle

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 733

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

Open AccessArticle

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

Cited by 1 | Viewed by 911

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

Open AccessArticle

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 966

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

Open AccessEditor’s ChoiceArticle

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 7 | Viewed by 978

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

Open AccessArticle

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 8 | Viewed by 1008

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

Open AccessArticle

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 761

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

Open AccessArticle

On Sub Convexlike Optimization Problems

by Renying Zeng

Mathematics 2023, 11(13), 2928; https://doi.org/10.3390/math11132928 - 29 Jun 2023

Cited by 1 | Viewed by 879

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

Open AccessArticle

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 5 | Viewed by 1214

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

Open AccessArticle

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 1382

Abstract

In this paper, we prove the Hermite–Hadamard–Fejér type inequalities for coordinated

(h_{1}, h_{2})

-convex functions on the rectangle from the plane

R^{2}

. 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

(h_{1}, h_{2})

-convex functions on the rectangle from the plane

R^{2}

. 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

(h_{1}, h_{2})

-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

Open AccessArticle

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 1431

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

Open AccessArticle

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 1305

Abstract

In this paper, we present some new and novel mappings defined over

[0, 1]

with the help of

G A

-convex functions. As a consequence, we obtain companions of Fejér-type inequalities for

G A

-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

G A

-convex functions. As a consequence, we obtain companions of Fejér-type inequalities for

G A

-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

Open AccessArticle

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 6 | Viewed by 1624

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

Open AccessArticle

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 14 | Viewed by 1953

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 = ⟨ ω_{c}, ω_{r} ⟩ = ⟨ \frac{\bar{ω} + \underset{̲}{ω}}{2}, \frac{\bar{ω} - \underset{̲}{ω}}{2} ⟩

. 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-

(h_{1}, h_{2})

-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

Open AccessArticle

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 1188

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

Open AccessArticle

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 3 | Viewed by 1397

Abstract

We consider a sequence

{(Z_{j})}_{j \geq 1}

of i.i.d. d-dimensional random vectors and for every

n \geq 1

consider the sample

S_{n} = {Z_{1}, Z_{2}, \dots, Z_{n}} .

We say that [...] Read more.

We consider a sequence

{(Z_{j})}_{j \geq 1}

of i.i.d. d-dimensional random vectors and for every

n \geq 1

consider the sample

S_{n} = {Z_{1}, Z_{2}, \dots, Z_{n}} .

We say that

Z_{j}

is a “leader” in the sample

S_{n}

if

Z_{j} \geq Z_{k},

\forall k \in {1, 2, \dots, n}

and

Z_{j}

is an “anti-leader” if

Z_{j} \leq Z_{k},

\forall k \in {1, 2, \dots, n}

. After all, the leader and the anti-leader are the naive extremes. Let

a_{n}

be the probability that

S_{n}

has a leader,

b_{n}

be the probability that

S_{n}

has an anti-leader and

c_{n}

be the probability that

S_{n}

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

{(Z_{j})}_{j \geq 1}

so that the inferior limits of

a_{n}, b_{n}, c_{n}

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

Open AccessArticle

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 9 | Viewed by 1657

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

Open AccessEditor’s ChoiceArticle

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 1919

Abstract

In this article, we give sharp two-sided bounds for the generalized Jensen functional

J_{n} (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

J_{n} (f, g, h;

p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean

A_{n} (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

Open AccessArticle

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 11 | Viewed by 1546

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

Open AccessArticle

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 12 | Viewed by 2214

Abstract

In this paper, using the notions of

q^{κ_{2}}

-quantum integral and

q^{κ_{2}}

-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

q^{κ_{2}}

-quantum integral and

q^{κ_{2}}

-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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