Assume that the function $f:[0,\infty )\rightarrow \mathbb {R} $ is operator monotone in $[0,\infty )$. We can define the perspective $\mathcal {P}_{f}\left (B,A\right )$ by setting $$\mathcal{P}_{f}\left( B,A\right) :=A^{1/2}f\left( A^{-1/2}BA^{-1/2}\right) A^{1/2}, $$ where $A, B>0$. In this paper, we show among others that, if $\sigma \geq C \geq \rho>0$, $D>0$, $\varsigma\geq Q\geq\tau>0$ and $0<$$n\leq D-C\leq N$ for some constants $\rho, \sigma, \varsigma, \tau, n, N$, then $$\begin{array}{@{}rcl@{}} 0& \le& \frac{n}{N{\varsigma}^{2}}\left[ \mathcal{P}_{f}\left( {\varsigma} ,N+\sigma \right) -\mathcal{P}_{f}\left( {\varsigma} ,\sigma \right) \right] Q^{2} \\ & \leq& \mathcal{P}_{f}\left( Q,D\right) -\mathcal{P}_{f}\left( Q,C\right) \\ & \leq& \frac{N}{n\tau^{2}}\left[ \mathcal{P}_{f}\left( \tau ,n+\rho \right) -\mathcal{P}_{f}\left( \tau ,\rho \right) \right] Q^{2}. \end{array} $$ Applications for the weighted operator geometric mean and the perspective $$\mathcal{P}_{\ln \left( \cdot +1\right) }\left( B,A\right) :=A^{1/2}\ln \left( A^{-1/2}BA^{-1/2}+1\right) A^{1/2},~ A,B>0 $$ are also provided.