The best constant in the Davis inequality for the expectation of the martingale square function

A method is introduced for the simultaneous study of the square function and the maximal function of a martingale that can yield sharp norm inequalities between the two. One application is that the expectation of the square function of a martingale is not greater than root3 times the expectation of the maximal function. This gives the best constant for one side of the Davis two-sided inequality. The martingale may take its values in any real or complex Hilbert space. The elementary discrete-time case leads quickly to the analogous results for local martingales M indexed by [0, infinity). Some earlier inequalities are also improved and, closely related, the Levy martingale is embedded in a large family of submartingales.