If $\left\{ E_{\lambda }\right\} _{\lambda \in \mathbb{R}}$ is the spectral family of the bounded selfadjoint operator $A$ on the Hilbert space $H$ and $ m=\min Sp\left( A\right) $ and $M=\max Sp\left( A\right)$, we show that for any continuous function $\varphi $: $\left[ m,M\right] \rightarrow \mathbb{C}$ we have the inequality \begin{eqnarray*} \left\vert \left\langle \varphi \left( A\right) x,y\right\rangle \right\vert ^{2} &\leq &\left( \int_{m-0}^{M}\left\vert \varphi \left( t\right) \right\vert d\left( \bigvee_{m-0}^{t}\left( \left\langle E_{\left( \cdot \right) }x,y\right\rangle \right) \right) \right) ^{2} \\ &\leq &\left\langle \left\vert \varphi \left( A\right) \right\vert x,x\right\rangle \left\langle \left\vert \varphi \left( A\right) \right\vert y,y\right\rangle \end{eqnarray*} for any vector $x$ and $y$ from $H.$ Some related results and applications are also given.