A sharp integral inequality for the dyadic maximal operator and a related stability result

We prove a sharp integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables is possible, as can be seen in Our inequality of interest is proved in this article by a simpler and more immediate way. We also study a stability result in connection with this inequality, that is we provide a necessary and sufficient condition, for a sequence of functions, under which we obtain equality in the limit. The proof of this result is based on the proof of the related inequality which we present in this article.