Mathematics

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14 pages, 352 KiB

Open AccessArticle

Generalized Lorentzian Sasakian-Space-Forms with

M

-Projective Curvature Tensor

by D. G. Prakasha, M. R. Amruthalakshmi, Fatemah Mofarreh and Abdul Haseeb

Mathematics 2022, 10(16), 2869; https://doi.org/10.3390/math10162869 - 11 Aug 2022

Cited by 3 | Viewed by 1259

Abstract

In this note, the generalized Lorentzian Sasakian-space-form

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

satisfying certain constraints on the

M

-projective curvature tensor

W^{*}

is considered. Here, we characterize the structure [...] Read more.

In this note, the generalized Lorentzian Sasakian-space-form

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

satisfying certain constraints on the

M

-projective curvature tensor

W^{*}

is considered. Here, we characterize the structure

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

when it is, respectively,

M

-projectively flat,

M

-projectively semisymmetric,

M

-projectively pseudosymmetric, and

φ - M

-projectively semisymmetric. Moreover,

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

satisfies the conditions

W^{*} (ζ, V_{1}) \cdot S = 0

,

W^{*} (ζ, V_{1}) \cdot R = 0

and

W^{*} (ζ, V_{1}) \cdot W^{*} = 0

are also examined. Finally, illustrative examples are given for obtained results. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

12 pages, 274 KiB

Open AccessEditor’s ChoiceArticle

Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

by Simona Decu and Stefan Haesen

Mathematics 2022, 10(3), 330; https://doi.org/10.3390/math10030330 - 21 Jan 2022

Cited by 7 | Viewed by 2836

Abstract

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely involving the Chen first invariant and the mean curvature of totally real and holomorphic spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Furthermore, we investigate the equality [...] Read more.

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely involving the Chen first invariant and the mean curvature of totally real and holomorphic spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Furthermore, we investigate the equality cases of these inequalities. As illustrations of the applications of the above inequalities, we consider a few examples. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

7 pages, 243 KiB

Open AccessArticle

On Characterizing a Three-Dimensional Sphere

by Nasser Bin Turki, Sharief Deshmukh and Gabriel-Eduard Vîlcu

Mathematics 2021, 9(24), 3311; https://doi.org/10.3390/math9243311 - 19 Dec 2021

Cited by 1 | Viewed by 1896

Abstract

In this paper, we find a characterization of the 3-sphere using 3-dimensional compact and simply connected trans-Sasakian manifolds of type

(α, β)

. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

15 pages, 295 KiB

Open AccessArticle

A Hardy–Hilbert-Type Inequality Involving Parameters Composed of a Pair of Weight Coefficients with Their Sums

by Bicheng Yang, Shanhe Wu and Xingshou Huang

Mathematics 2021, 9(22), 2950; https://doi.org/10.3390/math9222950 - 18 Nov 2021

Viewed by 1337

Abstract

In this paper, we establish a new Hardy–Hilbert-type inequality involving parameters composed of a pair of weight coefficients with their sum. Our result is a unified generalization of some Hardy–Hilbert-type inequalities presented in earlier papers. Based on the obtained inequality, the equivalent conditions [...] Read more.

In this paper, we establish a new Hardy–Hilbert-type inequality involving parameters composed of a pair of weight coefficients with their sum. Our result is a unified generalization of some Hardy–Hilbert-type inequalities presented in earlier papers. Based on the obtained inequality, the equivalent conditions of the best possible constant factor related to several parameters are discussed, and the equivalent forms and the operator expressions are also considered. As applications, we illustrate how the inequality obtained can generate some new Hardy–Hilbert-type inequalities. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

18 pages, 303 KiB

Open AccessArticle

On Golden Lorentzian Manifolds Equipped with Generalized Symmetric Metric Connection

by Majid Ali Choudhary, Khaled Mohamed Khedher, Oğuzhan Bahadır and Mohd Danish Siddiqi

Mathematics 2021, 9(19), 2430; https://doi.org/10.3390/math9192430 - 30 Sep 2021

Cited by 6 | Viewed by 1733

Abstract

This research deals with the generalized symmetric metric U-connection defined on golden Lorentzian manifolds. We also derive sharp geometric inequalities that involve generalized normalized

δ

-Casorati curvatures for submanifolds of golden Lorentzian manifolds equipped with generalized symmetric metric U-connection. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

23 pages, 362 KiB

Open AccessArticle

New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications

by Saima Rashid, Aasma Khalid, Omar Bazighifan and Georgia Irina Oros

Mathematics 2021, 9(15), 1753; https://doi.org/10.3390/math9151753 - 26 Jul 2021

Cited by 18 | Viewed by 1912

Abstract

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is [...] Read more.

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

9 pages, 235 KiB

Open AccessArticle

New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

by Shyam Sundar Santra, Omar Bazighifan and Mihai Postolache

Mathematics 2021, 9(11), 1159; https://doi.org/10.3390/math9111159 - 21 May 2021

Cited by 22 | Viewed by 1881

Abstract

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In [...] Read more.

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

17 pages, 300 KiB

Open AccessArticle

Sharp Inequalities for the Hardy–Littlewood Maximal Operator on Finite Directed Graphs

by Xiao Zhang, Feng Liu and Huiyun Zhang

Mathematics 2021, 9(9), 946; https://doi.org/10.3390/math9090946 - 23 Apr 2021

Cited by 1 | Viewed by 1489

Abstract

In this paper, we introduce and study the Hardy–Littlewood maximal operator

M_{\vec{G}}

on a finite directed graph

\vec{G}

. We obtain some optimal constants for the

ℓ^{p}

norm of

M_{\vec{G}}

by introducing two classes of directed graphs. [...] Read more.

In this paper, we introduce and study the Hardy–Littlewood maximal operator

M_{\vec{G}}

on a finite directed graph

\vec{G}

. We obtain some optimal constants for the

ℓ^{p}

norm of

M_{\vec{G}}

by introducing two classes of directed graphs. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

31 pages, 444 KiB

Open AccessArticle

Geometric Inequalities for Warped Products in Riemannian Manifolds

by Bang-Yen Chen and Adara M. Blaga

Mathematics 2021, 9(9), 923; https://doi.org/10.3390/math9090923 - 21 Apr 2021

Cited by 9 | Viewed by 3002

Abstract

Warped products are the most natural and fruitful generalization of Riemannian products. Such products play very important roles in differential geometry and in general relativity. After Bishop and O’Neill’s 1969 article, there have been many works done on warped products from intrinsic point [...] Read more.

Warped products are the most natural and fruitful generalization of Riemannian products. Such products play very important roles in differential geometry and in general relativity. After Bishop and O’Neill’s 1969 article, there have been many works done on warped products from intrinsic point of view during the last fifty years. In contrast, the study of warped products from extrinsic point of view was initiated around the beginning of this century by the first author in a series of his articles. In particular, he established an optimal inequality for an isometric immersion of a warped product

N_{1} \times_{f} N_{2}

into any Riemannian manifold

R^{m} (c)

of constant sectional curvature c which involves the Laplacian of the warping function f and the squared mean curvature

H^{2}

. Since then, the study of warped product submanifolds became an active research subject, and many papers have been published by various geometers. The purpose of this article is to provide a comprehensive survey on the study of warped product submanifolds which are closely related with this inequality, done during the last two decades. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

9 pages, 262 KiB

Open AccessArticle

Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold

by Amira Ishan, Sharief Deshmukh and Gabriel-Eduard Vîlcu

Mathematics 2021, 9(8), 863; https://doi.org/10.3390/math9080863 - 14 Apr 2021

Cited by 3 | Viewed by 2156

Abstract

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere

S^{m} (c)

of constant curvature c. The first characterization uses the well [...] Read more.

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere

S^{m} (c)

of constant curvature c. The first characterization uses the well known de-Rham Laplace operator, while the second uses a nontrivial solution of the famous Fischer–Marsden differential equation. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

24 pages, 352 KiB

Open AccessArticle

Geometry of Bi-Warped Product Submanifolds of Nearly Trans-Sasakian Manifolds

by Ali H. Alkhaldi and Akram Ali

Mathematics 2021, 9(8), 847; https://doi.org/10.3390/math9080847 - 13 Apr 2021

Cited by 2 | Viewed by 1642

Abstract

In the present work, we consider two types of bi-warped product submanifolds,

M = M_{T} \times_{f_{1}} M_{⊥} \times_{f_{2}} M_{ϕ}

and

M = M_{ϕ} \times_{f_{1}} M_{T} \times_{f_{2}} M_{⊥}

, in [...] Read more.

In the present work, we consider two types of bi-warped product submanifolds,

M = M_{T} \times_{f_{1}} M_{⊥} \times_{f_{2}} M_{ϕ}

and

M = M_{ϕ} \times_{f_{1}} M_{T} \times_{f_{2}} M_{⊥}

, in nearly trans-Sasakian manifolds and construct inequalities for the squared norm of the second fundamental form. The main results here are a generalization of several previous results. We also design some applications, in view of mathematical physics, and obtain relations between the second fundamental form and the Dirichlet energy. The relationship between the eigenvalues and the second fundamental form is also established. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

11 pages, 268 KiB

Open AccessArticle

Generalization of the Lieb–Thirring–Araki Inequality and Its Applications

by Yonggang Li, Jing Wang and Huafei Sun

Mathematics 2021, 9(7), 723; https://doi.org/10.3390/math9070723 - 26 Mar 2021

Viewed by 1868

Abstract

The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, [...] Read more.

The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, according to the properties of exterior algebra and the Schur-convex function, we provide a new proof for the generalization of the Lieb–Thirring–Araki theorem and Furuta theorem. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

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