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(2003 / vol.6 / no.2)
Estimations of Reverse Inequalities for Convex Functions footnotesize-- Operator Inequality derived from Quasi-Arithmetic Mean
Masaru Tominaga
Pages. 129-139     



We show some order relations between a generalized quasi-arithmetic mean and the arithmetic mean.
First we give an estimation $F(l)$ of the following difference:
$$gleft(<f(A)x,x>
ight) - l <Ax,x>$$
for $l>0$ and $|x|=1$, where $f(t)$, $g(t)$ are functions with some conditions and $A$ is a self-adjoint operator.
Next for self-adjoint operators $A_i$ we show order relations:
$$K_{f,g}(p) sum^n_{i=1}l_iA_i + M_{f,g}(p)
le gleft(sum^n_{i=1}l_if(A_i)
ight)
le L_{f,g}(q) sum^n_{i=1}l_iA_i + N_{f,g}(q)$$
under some conditions.
Moreover, we give ratio and difference inequalities from the above inequalities.
As applications, we show that some constants e.g. Ky Fan-Furuta constant, logarithmic mean and Specht ratio play an important role in their estimations.



1. Introduction
2. Main Theorems
3. Applications to some typical functions



quasi-arithmetic mean, arithmetic mean, Ky Fan-Furuta constant, Specht ratio, logarithmic mean



47A63