A Fr\"olicher-type inequality for generalized complex manifolds

We prove a Fr"olicher-type inequality for a compact generalized complex manifold $M$, and show that the equality holds if and only if $M$ satisfies the generalized $\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma. In particular, this gives a unified proof of analogous results in the complex and symplectic cases.