Let $P(I)$ be the set of all operator monotone functions defined on an interval $I$, and put $P_{+}(I)=\{h\in P(I):h(t)\geqq 0,h\ne 0\}$ and $P_{+}^{-1}(I)=\{h:h$ is increasing on $I,h^{-1}\in P_{+}(0,\infty )\}$. We will introduce a new set $L{P}_{+}(I)=\{h:h(t)>0$ on $I,logh\in P(I)\}$ and show $L{P}_{+}(I)·P_{+}^{-1}(I)\subset P_{+}^{-1}(I)$ for every right open interval $I$. By making use of this result, we will establish an operator inequality that generalizes simultaneously two well known operator inequalities. We will also show that if $p(t)$ is a real polynomial with a positive leading coefficient such that $p(0)=0$ and the other zeros of $p$ are all in $\{z:\mathrm{Rz}\leqq 0\}$ and if $q(t)$ is an arbitrary factor of $p(t)$, then $p(A{)}^2\leqq p(B{)}^2$ for $A,B\geqq 0$ implies $A^2\leqq B^2$ and $q(A{)}^2\leqq q(B{)}^2$.