Mathematics

Research

12 pages, 228 KiB

Open AccessArticle

On a Reverse Half-Discrete Hardy-Hilbert’s Inequality with Parameters

by Bicheng Yang, Shanhe Wu and Aizhen Wang

Mathematics 2019, 7(11), 1054; https://doi.org/10.3390/math7111054 - 4 Nov 2019

Cited by 27 | Viewed by 1775

By means of the weight functions, the idea of introduced parameters, and the Euler-Maclaurin summation formula, a reverse half-discrete Hardy-Hilbert’s inequality and the reverse equivalent forms are given. The equivalent statements of the best possible constant factor involving several parameters are considered. As [...] Read more.

By means of the weight functions, the idea of introduced parameters, and the Euler-Maclaurin summation formula, a reverse half-discrete Hardy-Hilbert’s inequality and the reverse equivalent forms are given. The equivalent statements of the best possible constant factor involving several parameters are considered. As applications, two results related to the case of the non-homogeneous kernel and some particular cases are obtained. Full article

(This article belongs to the Special Issue Inequalities)

6 pages, 244 KiB

Open AccessFeature PaperArticle

On the Generalization for Some Power-Exponential-Trigonometric Inequalities

by Aníbal Coronel, Peter Kórus, Esperanza Lozada and Elias Irazoqui

Mathematics 2019, 7(10), 988; https://doi.org/10.3390/math7100988 - 17 Oct 2019

Cited by 1 | Viewed by 2215

Abstract

In this paper, we introduce and prove several generalized algebraic-trigonometric inequalities by considering negative exponents in the inequalities. Full article

(This article belongs to the Special Issue Inequalities)

19 pages, 357 KiB

Open AccessArticle

Variation Inequalities for One-Sided Singular Integrals and Related Commutators

by Feng Liu, Seongtae Jhang, Sung-Kwun Oh and Zunwei Fu

Mathematics 2019, 7(10), 876; https://doi.org/10.3390/math7100876 - 20 Sep 2019

Cited by 3 | Viewed by 2183

Abstract

We establish one-sided weighted endpoint estimates for the

ϱ

-variation (

ϱ > 2

) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the

ϱ

-variation operators of [...] Read more.

We establish one-sided weighted endpoint estimates for the

ϱ

-variation (

ϱ > 2

) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the

ϱ

-variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the

λ

-jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones. Full article

(This article belongs to the Special Issue Inequalities)

12 pages, 742 KiB

Open AccessArticle

New Inequalities of Weaving K-Frames in Subspaces

by Zhong-Qi Xiang

Mathematics 2019, 7(9), 863; https://doi.org/10.3390/math7090863 - 18 Sep 2019

Viewed by 2082

Abstract

In the present paper, we obtain some new inequalities for weaving K-frames in subspaces based on the operator methods. The inequalities are associated with a sequence of bounded complex numbers and a parameter

λ \in R

. We also give a double [...] Read more.

In the present paper, we obtain some new inequalities for weaving K-frames in subspaces based on the operator methods. The inequalities are associated with a sequence of bounded complex numbers and a parameter

λ \in R

. We also give a double inequality for weaving K-frames with the help of two bounded linear operators induced by K-dual. Facts prove that our results cover those recently obtained on weaving frames due to Li and Leng, and Xiang. Full article

(This article belongs to the Special Issue Inequalities)

11 pages, 253 KiB

Open AccessArticle

Difference Mappings Associated with Nonsymmetric Monotone Types of Fejér’s Inequality

by Mohsen Rostamian Delavar and Manuel De La Sen

Mathematics 2019, 7(9), 802; https://doi.org/10.3390/math7090802 - 1 Sep 2019

Cited by 1 | Viewed by 1512

Abstract

Two mappings

L_{w}

and

P_{w}

, in connection with Fejér’s inequality, are considered for the convex and nonsymmetric monotone functions. Some basic properties and results along with some refinements for Fejér’s inequality according to these new settings are obtained. As applications, [...] Read more.

Two mappings

L_{w}

and

P_{w}

, in connection with Fejér’s inequality, are considered for the convex and nonsymmetric monotone functions. Some basic properties and results along with some refinements for Fejér’s inequality according to these new settings are obtained. As applications, some special means type inequalities are given. Full article

(This article belongs to the Special Issue Inequalities)

15 pages, 310 KiB

Open AccessArticle

A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative

by Jehad Alzabut, Weerawat Sudsutad, Zeynep Kayar and Hamid Baghani

Mathematics 2019, 7(8), 747; https://doi.org/10.3390/math7080747 - 15 Aug 2019

Cited by 10 | Viewed by 3350

Abstract

New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their [...] Read more.

New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their features. We prove our main result in light of some efficient comparison analyses. The Gronwall–Bellman inequality in the case of weighted function is also obtained. By the help of the new proposed inequalities, examples of Riemann–Liouville and Caputo proportional fractional initial value problems are presented to emphasize the solution dependence on the initial data and on the right-hand side. Full article

(This article belongs to the Special Issue Inequalities)

7 pages, 245 KiB

Open AccessArticle

Refinements of Majorization Inequality Involving Convex Functions via Taylor’s Theorem with Mean Value form of the Remainder

by Shanhe Wu, Muhammad Adil Khan and Hidayat Ullah Haleemzai

Mathematics 2019, 7(8), 663; https://doi.org/10.3390/math7080663 - 24 Jul 2019

Cited by 9 | Viewed by 2907

Abstract

The aim of this paper is to establish some refined versions of majorization inequality involving twice differentiable convex functions by using Taylor theorem with mean-value form of the remainder. Our results improve several results obtained in earlier literatures. As an application, the result [...] Read more.

The aim of this paper is to establish some refined versions of majorization inequality involving twice differentiable convex functions by using Taylor theorem with mean-value form of the remainder. Our results improve several results obtained in earlier literatures. As an application, the result is used for deriving a new fractional inequality. Full article

(This article belongs to the Special Issue Inequalities)

9 pages, 242 KiB

Open AccessArticle

Hermite-Hadamard Type Inequalities for Interval (h₁, h₂)-Convex Functions

by Yanrong An, Guoju Ye, Dafang Zhao and Wei Liu

Mathematics 2019, 7(5), 436; https://doi.org/10.3390/math7050436 - 17 May 2019

Cited by 48 | Viewed by 2911

Abstract

We introduce the concept of interval

(h_{1}, h_{2})

-convex functions. Under the new concept, we establish some new interval Hermite-Hadamard type inequalities, which generalize those in the literature. Also, we give some interesting examples. [...] Read more.

We introduce the concept of interval

(h_{1}, h_{2})

-convex functions. Under the new concept, we establish some new interval Hermite-Hadamard type inequalities, which generalize those in the literature. Also, we give some interesting examples. Full article

(This article belongs to the Special Issue Inequalities)

18 pages, 316 KiB

Open AccessArticle

Generalized Steffensen’s Inequality by Fink’s Identity

by Asfand Fahad, Saad Ihsan Butt and Josip Pečarić

Mathematics 2019, 7(4), 329; https://doi.org/10.3390/math7040329 - 4 Apr 2019

Cited by 1 | Viewed by 2738

Abstract

By using Fink’s Identity, Green functions, and Montgomery identities we prove some identities related to Steffensen’s inequality. Under the assumptions of n-convexity and n-concavity, we give new generalizations of Steffensen’s inequality and its reverse. Generalizations of some inequalities (and their reverse), [...] Read more.

By using Fink’s Identity, Green functions, and Montgomery identities we prove some identities related to Steffensen’s inequality. Under the assumptions of n-convexity and n-concavity, we give new generalizations of Steffensen’s inequality and its reverse. Generalizations of some inequalities (and their reverse), which are related to Hardy-type inequality. New bounds of Gr

\ddot{u}

ss and Ostrowski-type inequalities have been proved. Moreover, we formulate generalized Steffensen’s-type linear functionals and prove their monotonicity for the generalized class of

(n + 1)

-convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions. Full article

(This article belongs to the Special Issue Inequalities)

14 pages, 254 KiB

Open AccessEditor’s ChoiceArticle

New Integral Inequalities via the Katugampola Fractional Integrals for Functions Whose Second Derivatives Are Strongly η-Convex

by Seth Kermausuor, Eze R. Nwaeze and Ana M. Tameru

Mathematics 2019, 7(2), 183; https://doi.org/10.3390/math7020183 - 15 Feb 2019

Cited by 17 | Viewed by 2963

Abstract

In this paper, we introduced some new integral inequalities of the Hermite–Hadamard type for functions whose second derivatives in absolute values at certain powers are strongly

η

-convex functions via the Katugampola fractional integrals. Full article

(This article belongs to the Special Issue Inequalities)

18 pages, 298 KiB

Open AccessArticle

Some Quantum Estimates of Hermite-Hadamard Inequalities for Quasi-Convex Functions

by Hefeng Zhuang, Wenjun Liu and Jaekeun Park

Mathematics 2019, 7(2), 152; https://doi.org/10.3390/math7020152 - 5 Feb 2019

Cited by 44 | Viewed by 2639

Abstract

In this paper, we develop some quantum estimates of Hermite-Hadamard type inequalities for quasi-convex functions. In some special cases, these quantum estimates reduce to the known results. Full article

(This article belongs to the Special Issue Inequalities)

11 pages, 758 KiB

Open AccessArticle

More on Inequalities for Weaving Frames in Hilbert Spaces

by Zhong-Qi Xiang

Mathematics 2019, 7(2), 141; https://doi.org/10.3390/math7020141 - 2 Feb 2019

Cited by 5 | Viewed by 2367

Abstract

In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real [...] Read more.

In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real numbers. It is indicated that our results are more general and cover the corresponding results recently obtained by Li and Leng. We also give a triangle inequality for weaving frames in Hilbert spaces, which is structurally different from previous ones. Full article

(This article belongs to the Special Issue Inequalities)

7 pages, 223 KiB

Open AccessArticle

New Refinement of the Operator Kantorovich Inequality

by Hamid Reza Moradi, Shigeru Furuichi and Zahra Heydarbeygi

Mathematics 2019, 7(2), 139; https://doi.org/10.3390/math7020139 - 1 Feb 2019

Viewed by 2652

Abstract

We focus on the improvement of operator Kantorovich type inequalities. Among the consequences, we improve the main result of the paper [H.R. Moradi, I.H. Gümüş, Z. Heydarbeygi, A glimpse at the operator Kantorovich inequality, Linear Multilinear Algebra, doi:10.1080/03081087.2018.1441799]. Full article

(This article belongs to the Special Issue Inequalities)

11 pages, 273 KiB

Open AccessArticle

Coefficient Inequalities of Functions Associated with Hyperbolic Domains

by Sarfraz Nawaz Malik, Shahid Mahmood, Mohsan Raza, Sumbal Farman, Saira Zainab and Nazeer Muhammad

Mathematics 2019, 7(1), 88; https://doi.org/10.3390/math7010088 - 16 Jan 2019

Cited by 4 | Viewed by 2601

Abstract

In this work, our focus is to study the Fekete-Szegö functional in a different and innovative manner, and to do this we find its upper bound for certain analytic functions which give hyperbolic regions as image domain. The upper bounds obtained in this [...] Read more.

In this work, our focus is to study the Fekete-Szegö functional in a different and innovative manner, and to do this we find its upper bound for certain analytic functions which give hyperbolic regions as image domain. The upper bounds obtained in this paper give refinement of already known results. Moreover, we extend our work by calculating similar problems for the inverse functions of these certain analytic functions for the sake of completeness. Full article

(This article belongs to the Special Issue Inequalities)

10 pages, 237 KiB

Open AccessArticle

Some Inequalities for g-Frames in Hilbert C*-Modules

by Zhong-Qi Xiang

Mathematics 2019, 7(1), 25; https://doi.org/10.3390/math7010025 - 27 Dec 2018

Viewed by 2309

Abstract

In this paper, we obtain new inequalities for g-frames in Hilbert

C^{*}

-modules by using operator theory methods, which are related to a scalar

λ \in R

and an adjointable operator with respect to two g-Bessel sequences. It is demonstrated that our [...] Read more.

In this paper, we obtain new inequalities for g-frames in Hilbert

C^{*}

-modules by using operator theory methods, which are related to a scalar

λ \in R

and an adjointable operator with respect to two g-Bessel sequences. It is demonstrated that our results can lead to several known results on this topic when suitable scalars and g-Bessel sequences are chosen. Full article

(This article belongs to the Special Issue Inequalities)

11 pages, 276 KiB

Open AccessArticle

Coefficient Inequalities of Functions Associated with Petal Type Domains

by Sarfraz Nawaz Malik, Shahid Mahmood, Mohsan Raza, Sumbal Farman and Saira Zainab

Mathematics 2018, 6(12), 298; https://doi.org/10.3390/math6120298 - 3 Dec 2018

Cited by 20 | Viewed by 2809

Abstract

In the theory of analytic and univalent functions, coefficients of functions’ Taylor series representation and their related functional inequalities are of major interest and how they estimate functions’ growth in their specified domains. One of the important and useful functional inequalities is the [...] Read more.

In the theory of analytic and univalent functions, coefficients of functions’ Taylor series representation and their related functional inequalities are of major interest and how they estimate functions’ growth in their specified domains. One of the important and useful functional inequalities is the Fekete-Szegö inequality. In this work, we aim to analyze the Fekete-Szegö functional and to find its upper bound for certain analytic functions which give parabolic and petal type regions as image domains. Coefficient inequalities and the Fekete-Szegö inequality of inverse functions to these certain analytic functions are also established in this work. Full article

(This article belongs to the Special Issue Inequalities)

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