Rankin–Cohen brackets of Jacobi forms and Jacobi Poincaré series

We prove that if a Rankin–Cohen bracket of a Jacobi form $\phi $ and a complex-valued holomorphic function $\psi $ defined on $\mathcal H\times \mathbb C$ is a Jacobi form, then $\psi $ is a Jacobi form. We also study the relation between Rankin–Cohen brackets of Jacobi forms and Jacobi Poincaré series.