Let $E$ be a complex Hilbert space.

Let $(A_1,...,A_n) \in \mathcal{L}(E)^n$, could you please help me to show that $$\displaystyle\sup_{\|x\|=1}\bigg(\displaystyle\sum_{i=1}^n\| A_i^*x\|^2\bigg)\leq (4n)\displaystyle\sup_{\|x\|=1}\bigg(\displaystyle\sum_{i=1}^n|\langle A_ix\;,\;x\rangle|^2\bigg).$$

And you for you help.