Chapter 1. Introduction ......................................... 1
Chapter 2. Preliminaries ........................................ 7
Chapter 3. Fourier transforms and positive definiteness ........ 13
3.1. Sinh(αt) ............................................ 13
cosh t
3.2. cosh(αt)sinh(βt) ................................... 15
cosh t
3.3. Sinh(αt)cosh(βt) and Sinh(αt)sinh(βt) .............. 17
sinh t sinh t
3.4. cosh(Παt) and sinh(Παt) ........................ 19
cosh2(Πt/2) cosh2(Πt/2)
3.5. t cosh(αt) and t sinh(αt) .......................... 22
sinh t sinh t
3.6. t(cosh(αt) + β) ...................................... 23
sinh t
3.7. t(cosh(βt) + γ)sinh(αt) .............................. 27
sinh t
3.8. t(cosh(βt) + γ)cosh(αt) .............................. 31
cosh t
Chapter 4. A certain Heinz-type inequality and related
commutator estimates ................................ 37
4.1. A certain Heinz-type inequality and its consequences ... 37
4.2. Certain commutator estimates ........................... 40
Chapter 5. Norm comparison for various operator means .......... 43
5.1. Heinz and A-L-G interpolation means .................... 43
5.2. Heinz means and the binomial mean B1/2 .................. 46
Chapter 6. Norm inequalities for H1/2+βXK1/2-β + H1/2-βXK1/2+β ±
H1/2XK1/2 ............................................ 49
6.1. Comparison to the logarithmic mean ..................... 49
6.2. Comparison to A-L-G interpolation means ................ 50
Chapter 7. Norm comparison of Heron-type means and related
topics .............................................. 55
7.1. Criterion for positive definiteness .................... 55
7.2. Heron-type means and a one-parameter version of Heinz
inequality ............................................. 59
7.3. Miscellaneous results .................................. 61
Chapter 8. Operator Lehmer means and their properties .......... 65
Appendix A. A direct proof for Proposition 7.3 ................. 69
Appendix B. Proof for Theorem 7.10 ............................. 73
Bibliography ................................................... 77
Index .......................................................... 79
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