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 $m=minSp\left(A\right)$ and $M=maxSp\left(A\right)$, we show that for any continuous function $\phi$: $[m,M]\to \mathbb{C}$ we have the inequality $$\begin{array}{rcl}{\left|⟨\phi \left(A\right)x,y⟩\right|}^2& \le & {\left({\int }_{m-0}^M\left|\phi \left(t\right)\right|d\left(\underset{m-0}{\overset{t}{\bigvee }}\left(⟨E_{(\cdot )}x,y⟩\right)\right)\right)}^2\\ & \le & ⟨\left|\phi \left(A\right)\right|x,x⟩⟨\left|\phi \left(A\right)\right|y,y⟩\end{array}$$ for any vector $x$ and $y$ from $H.$ Some related results and applications are also given.