Let $A$ and $B$ be positive operators on a Hilbert space. 
We consider what kind of conditions induces the order 
between $A$ and $B$. Such an attempt was done in recent 
works due to Yang and Ito. Based on their results, we 
prove that if $A^t\ \natural_{\frac{\gamma-t}{p-t}}\ 
B^p \ge B^{\gamma}$ for $0 < p < t$ and $p < \gamma$, 
then $A^{\beta} \ge B^{\beta}$ for $\beta = 
\min\{\gamma, t\}$, where the binary operation $\natural$ 
is used as a generalized formula of the geometric mean. 
Moreover we give an extension of Ito's theorem.