Let $\mathbf{P}(I)$ be the set of all operator monotone functions defined on an interval $I$ , and put $\mathbf{P}_{+}(I)=\{h\in \mathbf{P}(I): h(t)\geqq 0, h\neq 0\}$ and $\mathbf{P}_{+}^{-1}(I) = \{h: h$ is increasing on $I, h^{-1}\in \mathbf{P}_{+}(0,\infty)\}$ . We will introduce a new set $\mathbf{L}\mathbf{P}_{+}(I)=\{h:h(t)>0$ on $I, \log h \in \mathbf{P}(I)\}$ and show $\mathbf{L}\mathbf{P}_{+}(I)\cdot \mathbf{P}_{+}^{-1}(I)\subset \mathbf{P}_{+}^{-1}(I)$ 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 $\{z:Rz\leqq 0\}$ 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 $q(A)^{2}\leqq q(B)^{2}$ .