Sharp integrability exponents and constants for Muckenhoupt and Gehring weights as solution to a unique equation

The sharp results for the self-improving and the transition properties of Gehring RHq and Muckenhoupt Ap weights are unified and improved into corresponding sharp results for weights satisfying a general reverse Hölder inequality. We show that the optimal exponents of integrability as well as the best constants in the integral inequalities can be obtained by mean of the unique algebraic equation (x/x - q)1/q = B (x/x - p)1/p holding for the so called Bpq class (see 1.10) which contains the Gehring and Muckenhoupt classes as particular cases.