A Derivative-Hilbert operator acting on BMOA space

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix ${\mathcal{H}}_{\mu }=({\mu }_{n,k}{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k}={\mu }_{n+k}$, where ${\mu }_n={\int }_{[0,1)}{t}^{n}d\mu (t)$, induces, formally, the Derivative-Hilbert operator $${\mathcal{D}\mathcal{H}}_{\mu }(f)(z)=\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^{\mathrm{\infty }}{\mu }_{n,k}{a}_k\right)(n+1)z^n,z\in \mathbb{D},$$ where $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ is an analytic function in $\mathbb{D}$. We characterize the measures $\mu$ for which ${\mathcal{D}\mathcal{H}}_{\mu }$ is a bounded operator on $BMOA$ space. We also study the analogous problem from the $\alpha$-Bloch space ${\mathcal{B}}_{\alpha }(\alpha >0)$ into the $BMOA$ space.