The principal problem considered is the determination of all nonnegative functions, , for which there is a constant, , such that
where , is a fixed interval, is independent of , and is the Hardy maximal function,
The main result is that is such a function if and only if
where is any subinterval of , denotes the length of and is a constant independent of . Various related problems are also considered. These include weak type results, the problem when there are different weight functions on the two sides of the inequality, the case when or , a weighted definition of the maximal function, and the result in higher dimensions. Applications of the results to mean summability of Fourier and Gegenbauer series are also given.