Symmetry

Research

11 pages, 304 KiB

Open AccessArticle

A New Comprehensive Subclass of Analytic Bi-Univalent Functions Related to Gegenbauer Polynomials

by Tariq Al-Hawary, Ala Amourah, Abdullah Alsoboh and Omar Alsalhi

Symmetry 2023, 15(3), 576; https://doi.org/10.3390/sym15030576 - 22 Feb 2023

Cited by 12 | Viewed by 1443

In the current study, we provide a novel qualitative new subclass of analytical and bi-univalent functions in the symmetry domain

U

defined by the use of Gegenbauer polynomials. We derive estimates for the Fekete–Szegö functional problems and the Taylor–Maclaurin coefficients

|a_{2}|

and [...] Read more.

In the current study, we provide a novel qualitative new subclass of analytical and bi-univalent functions in the symmetry domain

U

defined by the use of Gegenbauer polynomials. We derive estimates for the Fekete–Szegö functional problems and the Taylor–Maclaurin coefficients

|a_{2}|

and

|a_{3}|

for the functions that belong to each of these new subclasses of the bi-univalent function classes. Some more results are revealed after concentrating on the parameters employed in our main results. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

20 pages, 357 KiB

Open AccessArticle

On Some Generalizations of Cauchy–Schwarz Inequalities and Their Applications

by Najla Altwaijry, Kais Feki and Nicuşor Minculete

Symmetry 2023, 15(2), 304; https://doi.org/10.3390/sym15020304 - 21 Jan 2023

Cited by 10 | Viewed by 1708

Abstract

The aim of this paper is to provide new upper bounds of

ω (T)

, which denotes the numerical radius of a bounded operator T on a Hilbert space

(H, ⟨ \cdot, \cdot ⟩)

. We show [...] Read more.

The aim of this paper is to provide new upper bounds of

ω (T)

, which denotes the numerical radius of a bounded operator T on a Hilbert space

(H, ⟨ \cdot, \cdot ⟩)

. We show the Aczél inequality in terms of the operator

| T |

. Next, we give certain inequalities about the A-numerical radius

ω_{A} (T)

and the A-operator seminorm

{∥ T ∥}_{A}

of an operator T. We also present several results related to the

A

-numerical radius of

2 \times 2

block matrices of semi-Hilbert space operators, by using symmetric

2 \times 2

block matrices. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

14 pages, 305 KiB

Open AccessArticle

A Study on the Modified Form of Riemann-Type Fractional Inequalities via Convex Functions and Related Applications

by Muhammad Samraiz, Maria Malik, Kanwal Saeed, Saima Naheed, Sina Etemad, Manuel De la Sen and Shahram Rezapour

Symmetry 2022, 14(12), 2682; https://doi.org/10.3390/sym14122682 - 19 Dec 2022

Cited by 1 | Viewed by 1458

Abstract

In this article, we provide constraints for the sum by employing a generalized modified form of fractional integrals of Riemann-type via convex functions. The mean fractional inequalities for functions with convex absolute value derivatives are discovered. Hermite–Hadamard-type fractional inequalities for a symmetric convex [...] Read more.

In this article, we provide constraints for the sum by employing a generalized modified form of fractional integrals of Riemann-type via convex functions. The mean fractional inequalities for functions with convex absolute value derivatives are discovered. Hermite–Hadamard-type fractional inequalities for a symmetric convex function are explored. These results are achieved using a fresh and innovative methodology for the modified form of generalized fractional integrals. Some applications for the results explored in the paper are briefly reviewed. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

18 pages, 310 KiB

Open AccessArticle

Some Companions of Fejér-Type Inequalities for Harmonically Convex Functions

by Muhammad Amer Latif

Symmetry 2022, 14(11), 2268; https://doi.org/10.3390/sym14112268 - 28 Oct 2022

Cited by 4 | Viewed by 927

Abstract

In this paper, we present some mappings defined over

[0, 1]

related to the Fejér-type inequalities that have been established for harmonically convex functions. As a consequence, we obtain companions of Fejér-type inequalities for harmonically convex functions by using these mappings. Properties [...] Read more.

In this paper, we present some mappings defined over

[0, 1]

related to the Fejér-type inequalities that have been established for harmonically convex functions. As a consequence, we obtain companions of Fejér-type inequalities for harmonically convex functions by using these mappings. Properties of these mappings are discussed, and consequently, we obtain refinement inequalities of some known results. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

11 pages, 296 KiB

Open AccessArticle

Bounds and Completely Monotonicity of Some Functions Involving the Functions ψ′(l) and ψ″(l)

by Omelsaad Ahfaf, Ahmed Talat and Mansour Mahmoud

Symmetry 2022, 14(7), 1420; https://doi.org/10.3390/sym14071420 - 11 Jul 2022

Cited by 1 | Viewed by 1008

Abstract

Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In the paper, we prove the completely monotonic (CM) property of [...] Read more.

Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In the paper, we prove the completely monotonic (CM) property of some functions involving the function

Δ (l) = ψ^{″} (l) + {[ψ^{'} (l)]}^{2}

and hence we deduce a new double inequality for it. Additionally, we study the CM degree of some functions involving the function

ψ^{'} (l)

. Our new bounds takes priority over some of the recently published results. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

13 pages, 287 KiB

Open AccessArticle

Some Rational Approximations and Bounds for Bateman’s G-Function

by Omelsaad Ahfaf, Mansour Mahmoud and Ahmed Talat

Symmetry 2022, 14(5), 929; https://doi.org/10.3390/sym14050929 - 2 May 2022

Cited by 1 | Viewed by 1124

Abstract

Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In this paper, we present the following rational approximations for Bateman’s [...] Read more.

Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In this paper, we present the following rational approximations for Bateman’s G-function

G (w) = \frac{1}{w} + {[2 w^{2} + \sum_{j = 1}^{n} 4 α_{j} w^{2 - 2 j}]}^{- 1} + O (\frac{1}{w^{2 n + 2}}),

where

α_{1} = \frac{1}{4},

and

α_{j} = \frac{(1 - 2^{2 j + 2}) B_{2 j + 2}}{j + 1} + \sum_{ν = 1}^{j - 1} \frac{(1 - 2^{2 j - 2 ν + 2}) B_{2 j - 2 ν + 2} α_{ν}}{j - ν + 1}, j > 1 .

As a consequence, we introduced some new bounds of

G (w)

and a completely monotonic function involving it. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

16 pages, 335 KiB

Open AccessArticle

Hardy–Leindler, Yang and Hwang Inequalities for Functions of Several Variables via Time Scale Calculus

by Ammara Nosheen, Huma Akbar, Maroof Ahmad Sultan, Jae Dong Chung and Nehad Ali Shah

Symmetry 2022, 14(4), 802; https://doi.org/10.3390/sym14040802 - 12 Apr 2022

Viewed by 1131

Abstract

In this paper, Hardy–Leindler, Hardy–Yang and Hwang type inequalities are extended on time scales calculus. These extensions are depending upon use of symmetric multiple delta integrals. The target is achieved by utilizing some inequalities in literature along with mathematical induction principle and Fubini’s [...] Read more.

In this paper, Hardy–Leindler, Hardy–Yang and Hwang type inequalities are extended on time scales calculus. These extensions are depending upon use of symmetric multiple delta integrals. The target is achieved by utilizing some inequalities in literature along with mathematical induction principle and Fubini’s theorem on time scales. The obtained inequalities are discussed in discrete, continuous and quantum calculus in search of applications. Particular cases of proved results include Hardy, Copson, Hardy–Littlewood, Levinson and Bennett-type inequalities for symmetric sums. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

17 pages, 308 KiB

Open AccessArticle

Trapezium-like Inequalities Involving k-th Order Differentiable

R_{γ}

-Convex Functions and Applications

by Miguel Vivas-Cortez, Muhammad Uzair Awan, Sadia Talib, Muhammad Aslam Noor and Khalida Inayat Noor

Symmetry 2022, 14(3), 448; https://doi.org/10.3390/sym14030448 - 23 Feb 2022

Cited by 1 | Viewed by 1249

Abstract

We introduce the class of

R_{γ}

-convex functions and discuss that it relates to some other classes of convexity. We study the class of

R_{γ}

-convex functions in the perspective of trapezium-like inequalities, for which we also derive a new integral [...] Read more.

We introduce the class of

R_{γ}

-convex functions and discuss that it relates to some other classes of convexity. We study the class of

R_{γ}

-convex functions in the perspective of trapezium-like inequalities, for which we also derive a new integral identity involving a

k

-th order differentiable function. In order to show the significance of our results, we also discuss several special cases and offer some interesting applications. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

28 pages, 355 KiB

Open AccessArticle

New “Conticrete” Hermite–Hadamard–Jensen–Mercer Fractional Inequalities

by Shah Faisal, Muhammad Adil Khan, Tahir Ullah Khan, Tareq Saeed, Ahmed Mohammed Alshehri and Eze R. Nwaeze

Symmetry 2022, 14(2), 294; https://doi.org/10.3390/sym14020294 - 1 Feb 2022

Cited by 23 | Viewed by 1728

Abstract

The theory of symmetry has a significant influence in many research areas of mathematics. The class of symmetric functions has wide connections with other classes of functions. Among these, one is the class of convex functions, which has deep relations with the concept [...] Read more.

The theory of symmetry has a significant influence in many research areas of mathematics. The class of symmetric functions has wide connections with other classes of functions. Among these, one is the class of convex functions, which has deep relations with the concept of symmetry. In recent years, the Schur convexity, convex geometry, probability theory on convex sets, and Schur geometric and harmonic convexities of various symmetric functions have been extensively studied topics of research in inequalities. The present attempt provides novel portmanteauHermite–Hadamard–Jensen–Mercer-type inequalities for convex functions that unify continuous and discrete versions into single forms. They come as a result of using Riemann–Liouville fractional operators with the joint implementations of the notions of majorization theory and convex functions. The obtained inequalities are in compact forms, containing both weighted and unweighted results, where by fixing the parameters, new and old versions of the discrete and continuous inequalities are obtained. Moreover, some new identities are discovered, upon employing which, the bounds for the absolute difference of the two left-most and right-most sides of the main results are established. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

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