Generalized Pólya–Szegö type inequalities for some non-commutative geometric means

Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means

Copyright q 2010 B.-Y. Long and Y.-M. Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For p ∈ R, the generalized logarithmic mean L p a, b, arithmetic mean Aa, b, and geometric mean Ga, b of two positive numbers a and b are d...

Some Comparison Inequalities for Generalized Muirhead and Identric Means

Inequalities for Generalized Logarithmic Means

For p ∈ R, the generalized logarithmic mean Lp of two positive numbers a and b is defined as Lp a, b a, for a b, LP a, b b 1 − a 1 / p 1 b − a 1/p , for a/ b, p / − 1, p / 0, LP a, b b − a / log b − loga , for a/ b, p −1, and LP a, b 1/e b/a 1/ b−a , for a/ b, p 0. In this paper, we prove that G a, b H a, b 2L−7/2 a, b , A a, b H a, b 2L−2 a, b , and L−5 a, b H a, b for all a, b > 0, and the co...

On Hadamard Type Inequalities for Generalized Weighted Quasi-arithmetic Means

In the present paper we establish some integral inequalities analogous to the wellknown Hadamard inequality for a class of generalized weighted quasi-arithmetic means in integral form.

Non-commutative Martingale Inequalities

We prove the analogue of the classical Burkholder-Gundy inequalites for noncommutative martingales. As applications we give a characterization for an Ito-Clifford integral to be an L-martingale via its integrand, and then extend the Ito-Clifford integral theory in L, developed by Barnett, Streater and Wilde, to L for all 1 < p < ∞. We include an appendix on the non-commutative analogue of the c...