Paper's Title:

Ostrowski Type Inequalities for Lebesgue Integral: a Survey of Recent Results

Author(s):

Sever S. Dragomir^1,2

¹Mathematics, School of Engineering & Science
Victoria University, PO Box 14428
Melbourne City, MC 8001,
Australia
E-mail: sever.dragomir@vu.edu.au

 
²DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences,
School of Computer Science & Applied Mathematics,
University of the Witwatersrand,
Private Bag 3, Johannesburg 2050,
South Africa
URL: http://rgmia.org/dragomir 

Abstract:

The main aim of this survey is to present recent results concerning Ostrowski type inequalities for the Lebesgue integral of various classes of complex and real-valued functions. The survey is intended for use by both researchers in various fields of Classical and Modern Analysis and Mathematical Inequalities and their Applications, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas.

Paper's Title:

Positive Periodic Solutions for Second-Order Differential Equations with Generalized Neutral Operator

Author(s):

Wing-Sum Cheung, Jingli Ren and Weiwei Han

Abstract:

By some analysis of the neutral operator and an application of the fixed-point index theorem, we obtain sufficient conditions for the existence, multiplicity and nonexistence of periodic solutions to a second-order differential equation with the prescribed neutral operator, which improve and extend some recent results of Lu-Ge, Wu-Wang, and Zhang. An example is given to illustrate our results. Moreover, the analysis of the generalized neutral operator will be helpful for other types of differential equations.

Paper's Title:

Positive Solution For Discrete Three-Point Boundary Value Problems

Author(s):

Wing-Sum Cheung And Jingli Ren

Institute of Systems Science,
Chinese Academy of Sciences,
Beijing 100080, P.R. China
renjl@mx.amss.ac.cn

Abstract:

This paper is concerned with the existence of positive solution to the discrete three-point boundary value problem

,

where , and f is allowed to change sign. By constructing available operators, we shall apply the method of lower solution and the method of topology degree to obtain positive solution of the above problem for on a suitable interval. The associated Green’s function is first given.

Paper's Title:

Superquadracity, Bohr's Inequality and Deviation from a Mean Value

Author(s):

S. Abramovich, J. Barić, and J. Pečarić

Department of Mathematics, University of Haifa,
Haifa, 31905,
Israel
abramos@math.haifa.ac.il 

FESB, University of Split,
Rudera Bošcovića,
B.B., 21000, Split,
Croatia
jbaric@fesb.hr  

Faculty of Textile Technology, University of Zagreb,
Prilaz Baruna Filipovića,
30, 10000 Zagreb,
Croatia.
pecaric@hazu.hr  

Abstract:

Extensions of Bohr's inequality via superquadracity are obtained, where instead of the power p=2 which appears in Bohr's inequality we get similar results when we deal with p≥ 2 and with p≤ 2. Also, via superquadracity we extend a bound for deviation from a Mean Value.

Paper's Title:

Oscillation and Boundedness of Solutions to First and Second Order Forced Dynamic Equations with Mixed Nonlinearities

Author(s):

Ravi P. Agarwal and Martin Bohner

Department of Mathematical Sciences, Florida Institute of Technology
Melbourne, FL 32901,
U.S.A.
bohner@mst.edu
URL:http://web.mst.edu/~bohner

Department of Economics and Finance, Missouri University of Science and Technology
Rolla, MO 65401,
U.S.A.
agarwal@fit.edu

Abstract:

Some oscillation and boundedness criteria for solutions to certain first and second order forced dynamic equations with mixed nonlinearities are established. The main tool in the proofs is an inequality due to Hardy, Littlewood and Pólya. The obtained results can be applied to differential equations, difference equations and q-difference equations. The results are illustrated with numerous examples.

Paper's Title:

Regular Variation on Time Scales and Dynamic Equations

Author(s):

Pavel Řehák

Institute of Mathematics, Academy of Sciences of the Czech Republic
Žižkova 22, CZ61662 Brno,
Czech Republic
rehak@math.muni.cz
URL:http://www.math.muni.cz/~rehak

Abstract:

The purpose of this paper is twofold. First, we want to initiate a study of regular variation on time scales by introducing this concept in such a way that it unifies and extends well studied continuous and discrete cases. Some basic properties of regularly varying functions on time scales will be established as well. Second, we give conditions under which certain solutions of linear second order dynamic equations are regularly varying. Open problems and possible directions for a future research are discussed, too.

Paper's Title:

Inequalities for the Čebyšev Functional of Two Functions of Selfadjoint Operators in Hilbert Spaces

Author(s):

S. S. Dragomir

Abstract:

Some recent inequalities for the Čebyšev functional of two functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved functions and operators, are surveyed.

Paper's Title:

SEVERAL q-INTEGRAL INEQUALITIES

Author(s):

W. T. Sulaiman

Department of Computer Engineering,
College of Engineering,
University of Mosul,
Iraq.

 waadsulaiman@hotmail.com
 

Abstract:

In the present paper several q-integral inequalities are presented, some of them are new and others are generalizations of known results.

Paper's Title:

On Generalization of Hardy-type Inequalities

Author(s):

K. Rauf, S. Ponnusamy and J. O. Omolehin  

Department of Mathematics,
University of Ilorin, Ilorin,
Nigeria
krauf@unilorin.edu.ng

Department of Mathematics,
Indian Institute of Technology Madras,
Chennai- 600 036,
India
samy@iitm.ac.in

Department of Mathematics,
University of Ilorin, Ilorin,
Nigeria
omolehin_joseph@yahoo.com

Abstract:

This paper is devoted to some new generalization of Hardy-type integral inequalities and the reversed forms. The study is to determine conditions on which the generalized inequalities hold using some known hypothesis. Improvement of some inequalities are also presented.

Paper's Title:

A Geometric Generalization of Busemann-Petty Problem

Author(s):

Liu Rong and Yuan Jun

Shanghai Zhangjiang Group Junior Middle School,
Huo Xiang Road, Shanghai, 201203,
China

yuanjun_math@126.com

liurong@shu.edu.cn

Abstract:

The norm defined by Busemann's inequality establishes a class of star body - intersection body. This class of star body plays a key role in the solution of Busemann-Petty problem. In 2003, Giannapoulos [1] defined a norm for a new class of half-section. Based on this norm, we give a geometric generalization of Busemann-Petty problem, and get its answer as a result

Paper's Title:

On A Conjecture of A Logarithmically Completely Monotonic Function

Author(s):

Valmir Krasniqi, Armend Sh. Shabani

Department of Mathematics,
University of Prishtina,
Republic of Kosova

E-mail: vali.99@hotmail.com  
armend_shabani@hotmail.com

Abstract:

In this short note we prove a conjecture, related to a logarithmically completely monotonic function, presented in [5]. Then, we extend by proving a more generalized theorem. At the end we pose an open problem on a logarithmically completely monotonic function involving q-Digamma function.

Paper's Title:

Properties of q-gamma and q-beta functions derived from the q-Gauss-Pólya inequalities

Author(s):

Sanja Varošanec

Department of Mathematics,
University of Zagreb,
Zagreb, Croatia
E-mail: varosans@math.hr

Abstract:

We consider log-convexity and other properties of several functions related to q-gamma and q-beta functions. These properties are consequences of the general inequality, so-called q-analogue of the Gauss-Pólya inequality. Various inequalities involving these special functions are also given.

Paper's Title:

Inequalities for Functions of Selfadjoint Operators on Hilbert Spaces:
a Survey of Recent Results

Author(s):

Sever S. Dragomir^1,2

¹Mathematics, College of Engineering & Science
Victoria University, PO Box 14428
Melbourne City, MC 8001,
Australia
E-mail: sever.dragomir@vu.edu.au

 
²DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences,
School of Computer Science & Applied Mathematics,
University of the Witwatersrand,
Private Bag 3, Johannesburg 2050,
South Africa
URL: https://rgmia.org/dragomir 

Abstract:

The main aim of this survey is to present recent results concerning inequalities for continuous functions of selfadjoint operators on complex Hilbert spaces. It is intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas.

Paper's Title:

Trace Inequalities for Operators in Hilbert Spaces: a Survey of Recent Results

Author(s):

Sever S. Dragomir^1,2

¹Mathematics, School of Engineering & Science
Victoria University,
PO Box 14428 Melbourne City, MC 8001,
Australia
E-mail: sever.dragomir@vu.edu.au

 
²DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences,
School of Computer Science & Applied Mathematics,
University of the Witwatersrand,
Private Bag 3, Johannesburg 2050,
South Africa
URL: https://rgmia.org/dragomir 

Abstract:

In this paper we survey some recent trace inequalities for operators in Hilbert spaces that are connected to Schwarz's, Buzano's and Kato's inequalities and the reverses of Schwarz inequality known in the literature as Cassels' inequality and Shisha-Mond's inequality. Applications for some functionals that are naturally associated to some of these inequalities and for functions of operators defined by power series are given. Further, various trace inequalities for convex functions are presented including refinements of Jensen inequality and several reverses of Jensen's inequality. Hermite-Hadamard type inequalities and the trace version of Slater's inequality are given. Some Lipschitz type inequalities are also surveyed. Examples for fundamental functions such as the power, logarithmic, resolvent and exponential functions are provided as well.

Paper's Title:

On Some Nonlinear Retarded Integrodifferential Inequalities in Two and n Independent Variables and their Applications

Author(s):

Bitat Dalila and Khellaf Hassane

Department of Mathematics, Laboratory of Applied Mathematics and Modeling,
University of Constantine,
PO Box 325, Ain El Bey Road, Constantine 25017,
Algeria.
E-mail: bitat.dalila@umc.edu.dz khellafhassane@umc.edu.dz
URL: https://www.umc.edu.dz

Abstract:

In this paper, we establish some new nonlinear retarded integrodifferential inequalities in two and n independent variables. Some applications are given as illustration.

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