2016
1
1
1
146
Square inequality and strong order relation
2
2
It is wellknown that for Hilbert space linear operators $0 leq A$ and $0 leq C$, inequality$C leq A$ does not imply $C^2 leq A^2.$ We introduce a strong order relation $0 leq B lll A$, which guarantees that $C^2 leq B^{1/2}AB^{1/2} text{for all} 0 leq C leq B,$ and that $C^2 leq A^2$ when $B$ commutes with $A$. Connections of this approach with the arithmeticgeometric mean inequality of BhatiaKittaneh as well as the Kantorovich constant of $A$ are mentioned.
1

1
7


Tsuyoshi
Ando
Japan
ando@es.hokudai.ac.jp
Square inequality
strong order relation
operator arithmeticgeometric mean inequality
Kantorovich constant
Operators reversing orthogonality in normed spaces
2
2
We consider linear operators $Tcolon Xto X$ on a normed space $X$ which reverse orthogonality, i.e., satisfy the condition$$xbot yquad Longrightarrowquad Tybot Tx,qquad x,yin X,$$where $bot$ stands for Birkhoff orthogonality.
1

8
14


Jacek
Chmielinski
Pedagogical University of Cracow
Pedagogical University of Cracow
Poland
jacek@up.krakow.pl
Birkhoff orthogonality
orthogonality preserving mappings
orthogonality reversing mappings
linear similarities
characterizations of inner product spaces
Recent developments of Schwarz's type trace inequalities for operators in Hilbert spaces
2
2
In this paper, we survey some recent trace inequalities for operators inHilbert spaces that are connected to Schwarz's, Buzano's and Kato'sinequalities and the reverses of Schwarz inequality known in the literatureas Cassels' inequality and ShishaMond's inequality. Applications for somefunctionals that are naturally associated to some of these inequalities andfor functions of operators defined by power series are given. Examples forfundamental functions such as the power, logarithmic, resolvent andexponential functions are provided as well.
1

15
91


Sever
Dragomir
USA
sever.dragomir@vu.edu.au
trace class operators
HilbertSchmidt operators
Trace
Schwarz inequality
Kato inequality
Cassels inequality
ShishaMond inequality
Trace inequalities for matrices
Power series of operators
Fixed points of contractions and cyclic contractions on $C^{*}$algebravalued $b$metric spaces
2
2
In this paper, we discuss and improve some recent results aboutcontractive and cyclic mappings established in the framework of$C^{*}$algebravalued $b$metric spaces. Our proofs are muchshorter than the ones in existing literature. Also, we give twoexamples that support our approach.
1

92
103


Zoran
Kadelburg
Serbia
kadelbur@matf.bg.ac.rs


Antonella
Nastasi
Department of Mathematics and Computer Science,
University of Palermo
Department of Mathematics and Computer Science,
Un
Italy
ella.nastasi.93@gmail.com


Stojan
Radenovic
Faculty of Mechanical Engineering, University of Belgrade
Faculty of Mechanical Engineering, University
Serbia
radens@beotel.rs


Pasquale
Vetro
Department of Mathematics and Computer Science,
University of Palermo
Department of Mathematics and Computer Science,
Un
USA
pasquale.vetro@unipa.it
$C^{*}$algebravalued $b$metric space
$b$metric space
cyclic type mapping
expansive mapping
Strengthened converses of the Jensen and EdmundsonLahRibaric inequalities
2
2
In this paper, we give converses of the Jensen and EdmundsonLahRibaric inequalities which are more accurate than the existing ones. These converses are given in a difference form and they rely on the recent refinement of the Jensen inequality obtained via linear interpolation of a convex function. As an application, we also derive improved converse relations for generalized means, for the Holder and HermiteHadamard inequalities as well as for the inequalities of Giaccardi and Petrovic.
1

104
122


Mario
Krnic
USA
mario.krnic@fer.hr


Rozarija
Mikic
Croatia
jaksic.rozarija@gmail.com


Josip
Pecaric
USA
pecaric@element.hr
positive linear functional
convex function
converse
Jensen inequality
EdmundsonLahRibaric inequality
Holder inequality
HermiteHadamard inequality
Positive definite kernels and boundary spaces
2
2
We consider a kernel based harmonic analysis of "boundary,"and boundary representations. Our setting is general: certain classesof positive definite kernels. Our theorems extend (and are motivatedby) results and notions from classical harmonic analysis on the disk.Our positive definite kernels include those defined on infinite discretesets, for example sets of vertices in electrical networks, or discretesets which arise from sampling operations performed on positive definitekernels in a continuous setting. Below we give a summary of main conclusions in the paper: Startingwith a given positive definite kernel $K$ we make precise generalizedboundaries for $K$. They are measure theoretic "boundaries."Using the theory of Gaussian processes, we show that there is alwayssuch a generalized boundary for any positive definite kernel.
1

123
133


Feng
Tian
USA
james.ftian@gmail.com


Palle
Jorgensen
USA
pallejorgensen@uiowa.edu
Gaussian free fields
reproducing kernel Hilbert space
discrete analysis
Green's function
nonuniform sampling
(p,q)type beta functions of second kind
2
2
In the present article, we propose the (p,q)variant of beta function of second kind and establish a relation between the generalized beta and gamma functions using some identities of the postquantum calculus. As an application, we also propose the (p,q)BaskakovDurrmeyer operators, estimate moments and establish some direct results.
1

134
146


Ali
Aral
Turkey
aliaral73@yahoo.com


Vijay
Gupta
No
No
India
vijaygupta2001@hotmail.com
(p,q)beta function of second kind, (p
q)gamma function, Baskakov operator, Durrmeyer variant