15386

13790

كارشناسي ارشد

رياضي محض

اصفهان : دانشگاه صنعتي اصفهان

1398

[هفت],[78]ص.

محمود منجگاني

فريد بهرامي

واژه نامه

رسول نصر اصفهان، مهدي نعمتي

1398/10/15

كتابنامه

رياضي

رياضي

1398/10/15

2589627

INEQUALITIES RELATED TO HERON MEANS FOR POSITIVE OPERATORS FATEMEH HEIDARI fatemeh heidari@math iut ac ir December 2019 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Seyed Mahmoud Manjegani manjgani@cc iut ac irAdvisor Dr Farid Bahrami f bahrami@cc iut ac ir2000 MSC 47A63Keywords Positive operator Heron MeanAbstract In this thesis we first introduce some classical numerical inequalities and some refinements of them as well as showthat how these inequalities are generalized to operators on Hilbert space through Norm Trace Lowner order Fortwo positive real numbers a b and 0 1 the arithmetic geometric and harmonic means of a and bare defined respectively as follows a b 1 a b a b a 2 a 2 ba 2 a 2 1 1 1 1 a b 1 a 1 b 1 1 1In the case 2 we omit The following relation is well known a b a b a b The inequality a b a b is called arithmetic geometric mean inequality There are several means that interpolatebetween the basic three means One of them is the Heron mean which interpolates between the arithmetic mean andthe geometric mean and defined by Hr a b ra b 1 r a b The first two chapters of this thesis provide the preliminaries for the discussion of operator means inequalities InChapter 1 the necessary background about mean inequalities and their refinements for positive real numbers arepresented Chapter 2 introduces the aspects of singular values matrix norm and Lowner order that are needed forthe study of operator inequalities For two positive invertible operators A and B on a Hilbert space H the Heron mean is defined by Hr A B rA B 1 r A B In Chapter 3 Heron inequality is studied for matrices and this is continued in Chapter 4 for operators on infinitedimensional Hilbert spaces The final chapter discusses inequalities related to Heron means for positive operators The most important reasults in this thesis are as follows Theorem 1 Let A and B be two invertible positive matrices and r R Then the following inequalities arehold 1 If r 2 then rA B 1 r A B A B 2 If r 1 then rA B 1 r A B A B Theorem 2 Suppopse that A and B are two positive matrices and r R 1 If 0 1 and r 1 then Hr A B A B 2 2 2 If 0 1 and r r then Hr A B A B where r 3 1 Theorem 3 Let A and B be two positive matrices Then for 0 1 and r r we have Hr A B 1A B where r 1 4 1

محمود منجگاني

فريد بهرامي

رسول نصر اصفهان، مهدي نعمتي