Although the function $f(t)=t^p (p>1)$ is not operator monotone, Ando-Hiai 
showed an inequality $A^p \leq B^p$ on powers of positive operators $A$ and 
$B$ 
for a given $p>1$ under a condition $A \leq \phi_p(B)$ with a reasonable 
function $ \phi_p(t)$ instead of $A \leq B$.  On the other hand Furuta showed 
a 
similar inequality $A^p \leq \lambda _pB^p$ with a constant $\lambda_p$ under 
the condition $A \leq B$ and an additional one on $A$ or $B$, using convexity 
of 
the function $f(t)=t^p.$  In this paper, applying a convex inequality due to 
Mond-Pe\v{c}ari\'{c}, we extend the inequality of Furuta type to show an 
operator inequality on a functional order.