On the $\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma and Bott-Chern cohomology

On a compact complex manifold $X$, we prove a Fr"olicher-type inequality for Bott-Chern cohomology and we show that the equality holds if and only if $X$ satisfies the $\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma.