The best bound in the inequality of Hardy and Littlewood and its martingale counterpart
Proc. Amer. Math. Soc. 97 (1986), 429-436
Primary 26D15; Secondary 60G46
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Abstract: The smallest positive constant for which the Hardy and Littlewood  inequality (1) is valid is proved to be the unique positive solution of the equation . This settles a question raised by Dubins and Gilat (1978)  and, again, more recently, by D. Cox (1984) .
Numerically, . This should be compared with , obtained by Doob (1953)  in the context of martingale theory and, since then, widely used in the probability literature. Curiously enough, Doob's coefficient is the best upper bound, but for a slightly different inequality. If only the plus sign is removed from in (1), then must be at least for (1), so modified, to be valid. The original inequality (1) is a normalized form of the two-parameter inequality (2)
The set of all ordered pairs for which (2) holds is identified as Furthermore, for each point on the lower boundary of this set, there is a unique (up to null sets) which attains equality in (2).
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