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