The sharp constant in the reverse Hölder inequality for Muckenhoupt weights

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Bellman function for extremal problems in BMO (2012, background): “It is widely known that for the John–Nirenberg inequality and for the reverse Hölder inequality for the Ap-weights, the Bellman functions can be constructed similarly (compare [12, 13, 9] and [14, 11, 7]).”
Convex body domination and weighted estimates with matrix weights (2017, methodology): “The next fact is more subtle, it is proved in [17] by the Bellman function method and in [4] by a stopping time argument.”
Sharp results in the integral-form John-Nirenberg inequality (2007, methodology): “However, until [13] was published in 2003, none had found their Bellman functions explicitly, instead relying on Bellman-type arguments, when one uses a substitute function with required size and concavity properties.”