We prove a sharp integral inequality for the dyadic maximal function of φ ∈ L p . This inequality connects certain quantities related to integrals of φ and the dyadic maximal function of φ, under the hypothesis that the variables X φ dµ = f, X φ q dµ = A, 1 < q < p, are given, where 0 < f q ≤ A. Additionally, it contains a parameter β > 0 which when it attains a certain value depending only on f, A, q, the inequality becomes sharp. Using this inequality we give an alternative proof of the evaluation of the Bellman function related to the dyadic maximal operator of two integral variables.

We compute the Bellman function of three integral variables associated to the dyadic maximal operator on a subset of its domain. Additionally, we provide an upper bound for the whole domain of its definition.x ∈ R n :for every φ ∈ L 1 (R n ) and every λ > 0, and which is easily proved to be best possible. Further refinements of (1.2) can be seen in [10] and [11]. Then by using (1.2) and the well known Doob's method it is not difficult to prove that the following L p inequality is also true3) for every p > 1 and φ ∈ L p (R n ). Inequality (1.3) turns out to be best possible and its sharpness is proved in [17] (for general martingales see [1] and [2]).0

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