In the case of elliptic modular forms \textit{D. Zagier} [Modular functions of one variable VI, Lect. Notes Math. 627, 105-169 (1977; Zbl 0372.10017)] intensively investigated Rankin-Cohen differential operators. In the paper under review the authors study the analogous problem for Jacobi forms. They consider certain bilinear bracket operators \([ , ]_\nu\) on spaces of Jacobi forms, which involve the heat operator. The main aim is to compute the Petersson inner product of a Jacobi cusp form \(f\) against the bracket of a Jacobi modular form \(g\) and a Jacobi-Eisenstein series by Rankin's method. The result is a special value of some kind of Rankin convolution of \(f\) and \(g\), which for \(\nu>0\) has a somewhat different shape. Contrary to the one-variable situation, the arising series in general does not have multiplicative coefficients.