




Estimations of Reverse Inequalities for Convex Functions footnotesize Operator Inequality derived from QuasiArithmetic Mean 
Masaru Tominaga 
Pages. 129139 


We show some order relations between a generalized quasiarithmetic 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 selfadjoint operator.
Next for selfadjoint 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 FanFuruta constant, logarithmic mean and Specht ratio play an important role in their estimations. 

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


quasiarithmetic mean, arithmetic mean, Ky FanFuruta constant, Specht ratio, logarithmic mean 


47A63 





