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Axioms
Special Issues
Theory and Application of Integral Inequalities
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Theory and Application of Integral Inequalities

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 31 January 2025 | Viewed by 3706

Special Issue Editors

Dr. Loredana Ciurdariu

E-Mail Website
Guest Editor
Department of Mathematics, Politehnica University of Timisoara, 300006 Timisoara, Romania
Interests: mathematical inequalities; integral inequalities; quantum calculus; fractional calculus; dynamical systems
Prof. Dr. Eugenia Grecu

E-Mail Website
Guest Editor
Department of Management, Politehnica University of Timisoara, 300006 Timisoara, Romania
Interests: economics; dynamical systems; inequalities

Special Issue Information

Dear Colleagues,

Integral inequalities have gained an increasingly important role in many domains of research arising from pure mathematics and applied mathematics, captivating a higher interest of researchers now than in the previous decades. The integral inequalities are closely related with the concept of convexity.The aim of this Special Issue is to extend the inequalities obtained in the frame of q-calculus, fractional calculus and their further generalizations and to find the new types of integral inequalities for the different types of convexities for the better understanding and unification of these recently developed theories. The theory of variational inequalities is closely related to convex analysis. The optimality conditions of the differentiable convex functions are characterized by the variational inequalities.Integral inequalities and especially Jensen’s inequality have a special importance in optimization and information theory, statistics, cryptography and many other areas of research. Applications of integral inequalities in operator theory and matrix inequality would also be of interest in the various areas of pure mathematics.The guest editors would like to provide a platform to present the latest advances in the many aspects of the theory of integral inequalities along with their recently developed applications. 

Dr. Loredana Ciurdariu
Prof. Dr. Eugenia Grecu
Guest Editors

Manuscript Submission Information

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Keywords

  • generalized convexity
  • q-calculus
  • fractional calculus
  • variational inequalities
  • interval-valued inequalities
  • Jensen inequality
  • applications in information theory and statistics
  • inequalities related to functions
  • applications of inequalities in operator theory
  • matrix inequality
  • means

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Published Papers (6 papers)

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Research

21 pages, 688 KiB  
Article
Research on New Interval-Valued Fractional Integrals with Exponential Kernel and Their Applications
by Abdulrahman F. Aljohani, Ali Althobaiti and Saad Althobaiti
Axioms 2024, 13(9), 616; https://doi.org/10.3390/axioms13090616 - 11 Sep 2024
Viewed by 244
Abstract
This paper aims to introduce a new fractional extension of the interval Hermite–Hadamard (ℋℋ), ℋℋ–Fejér, and Pachpatte-type inequalities for left- and right-interval-valued harmonically convex mappings (ℒRIVH convex mappings) with an exponential function in the kernel. We use fractional operators to develop several [...] Read more.
This paper aims to introduce a new fractional extension of the interval Hermite–Hadamard (ℋℋ), ℋℋ–Fejér, and Pachpatte-type inequalities for left- and right-interval-valued harmonically convex mappings (ℒRIVH convex mappings) with an exponential function in the kernel. We use fractional operators to develop several generalizations, capturing unique outcomes that are currently under investigation, while also introducing a new operator. Generally, we propose two methods that, in conjunction with more generalized fractional integral operators with an exponential function in the kernel, can address certain novel generalizations of increasing mappings under the assumption of ℒRIV convexity, yielding some noteworthy results. The results produced by applying the suggested scheme show that the computational effects are extremely accurate, flexible, efficient, and simple to implement in order to explore the path of upcoming intricate waveform and circuit theory research. Full article
(This article belongs to the Special Issue Theory and Application of Integral Inequalities)
20 pages, 376 KiB  
Article
Geometric Characterization of Validity of the Lyapunov Convexity Theorem in the Plane for Two Controls under a Pointwise State Constraint
by Clara Carlota, Mário Lopes and António Ornelas
Axioms 2024, 13(9), 611; https://doi.org/10.3390/axioms13090611 - 9 Sep 2024
Viewed by 368
Abstract
This paper concerns control BVPs, driven by ODEs x′t=ut, using controls u0· &u1· in L1a,b,R2. We ask these two controls to satisfy a [...] Read more.
This paper concerns control BVPs, driven by ODEs x′t=ut, using controls u0· &u1· in L1a,b,R2. We ask these two controls to satisfy a very simple restriction: at points where their first coordinates coincide, also their second coordinates must coincide; which allows one to write (u1−u0)·=v·1,f· for some f·. Given a relaxed non bang-bang solution x¯·∈W1,1a,b,R2, a question relevant to applications was first posed three decades ago by A. Cellina: does there exist a bang-bang solution x^· having lower first-coordinate x^1·≤x¯1·? Being the answer always yes in dimension d=1, hence without f·, as proved by Amar and Cellina, for d=2 the problem is to find out which functions f· “are good”, namely “allow such 1-lower bang-bang solution x^· to exist”. The aim of this paper is to characterize “goodness of f·” geometrically, under “good data”. We do it so well that a simple computational app in a smartphone allows one to easily determine whether an explicitly given f· is good. For example: non-monotonic functions tend to be good; while, on the contrary, strictly monotonic functions are never good. Full article
(This article belongs to the Special Issue Theory and Application of Integral Inequalities)
31 pages, 1687 KiB  
Article
Some Classical Inequalities Associated with Generic Identity and Applications
by Muhammad Zakria Javed, Muhammad Uzair Awan, Bandar Bin-Mohsin, Hüseyin Budak and Silvestru Sever Dragomir
Axioms 2024, 13(8), 533; https://doi.org/10.3390/axioms13080533 - 6 Aug 2024
Viewed by 467
Abstract
In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s [...] Read more.
In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s inequality, Newton-type inequalities, and several two-point open trapezoidal inequalities. Also, we provide the numerical and visual explanation of our principal findings. Later, we provide some novel applications to the theory of means, special functions, error bounds of composite quadrature schemes, and parametric iterative schemes to find the roots of linear functions. Also, we attain several already known and new bounds for different values of γ and parameter ξ. Full article
(This article belongs to the Special Issue Theory and Application of Integral Inequalities)
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25 pages, 1378 KiB  
Article
Generalized Fuzzy-Valued Convexity with Ostrowski’s, and Hermite-Hadamard Type Inequalities over Inclusion Relations and Their Applications
by Miguel Vivas Cortez, Ali Althobaiti, Abdulrahman F. Aljohani and Saad Althobaiti
Axioms 2024, 13(7), 471; https://doi.org/10.3390/axioms13070471 - 12 Jul 2024
Viewed by 575
Abstract
Convex inequalities and fuzzy-valued calculus converge to form a comprehensive mathematical framework that can be employed to understand and analyze a broad spectrum of issues. This paper utilizes fuzzy Aumman’s integrals to establish integral inequalities of Hermite-Hahadard, Fejér, and Pachpatte types within up [...] Read more.
Convex inequalities and fuzzy-valued calculus converge to form a comprehensive mathematical framework that can be employed to understand and analyze a broad spectrum of issues. This paper utilizes fuzzy Aumman’s integrals to establish integral inequalities of Hermite-Hahadard, Fejér, and Pachpatte types within up and down (U·D) relations and over newly defined class U·D-ħ-Godunova–Levin convex fuzzy-number mappings. To demonstrate the unique properties of U·D-relations, recent findings have been developed using fuzzy Aumman’s, as well as various other fuzzy partial order relations that have notable deficiencies outlined in the literature. Several compelling examples were constructed to validate the derived results, and multiple notes were provided to illustrate, depending on the configuration, that this type of integral operator generalizes several previously documented conclusions. This endeavor can potentially advance mathematical theory, computational techniques, and applications across various fields. Full article
(This article belongs to the Special Issue Theory and Application of Integral Inequalities)
21 pages, 487 KiB  
Article
Hermite–Hadamard–Mercer-Type Inequalities for Three-Times Differentiable Functions
by Loredana Ciurdariu and Eugenia Grecu
Axioms 2024, 13(6), 413; https://doi.org/10.3390/axioms13060413 - 19 Jun 2024
Viewed by 501
Abstract
In this study, an integral identity is given in order to present some Hermite–Hadamard–Mercer-type inequalities for functions whose powers of the absolute values of the third derivatives are convex. Several consequences and three applications to special means are given, as well as four [...] Read more.
In this study, an integral identity is given in order to present some Hermite–Hadamard–Mercer-type inequalities for functions whose powers of the absolute values of the third derivatives are convex. Several consequences and three applications to special means are given, as well as four examples with graphics which illustrate the validity of the results. Moreover, several Hermite–Hadamard–Mercer-type inequalities for fractional integrals for functions whose powers of the absolute values of the third derivatives are convex are presented. Full article
(This article belongs to the Special Issue Theory and Application of Integral Inequalities)
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14 pages, 268 KiB  
Article
Weak Sharp Type Solutions for Some Variational Integral Inequalities
by Savin Treanţă and Tareq Saeed
Axioms 2024, 13(4), 225; https://doi.org/10.3390/axioms13040225 - 28 Mar 2024
Viewed by 834
Abstract
Weak sharp type solutions are analyzed for a variational integral inequality defined by a convex functional of the multiple integral type. A connection with the sufficiency property associated with the minimum principle is formulated, as well. Also, an illustrative numerical application is provided. [...] Read more.
Weak sharp type solutions are analyzed for a variational integral inequality defined by a convex functional of the multiple integral type. A connection with the sufficiency property associated with the minimum principle is formulated, as well. Also, an illustrative numerical application is provided. Full article
(This article belongs to the Special Issue Theory and Application of Integral Inequalities)
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