The best bound in the inequality of Hardy and Littlewood and its martingale counterpart
Author:
David Gilat
Journal:
Proc. Amer. Math. Soc. 97 (1986), 429436
MSC:
Primary 26D15; Secondary 60G46
MathSciNet review:
840624
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Abstract: The smallest positive constant for which the Hardy and Littlewood [8] 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) [6] and, again, more recently, by D. Cox (1984) [3]. Numerically, . This should be compared with , obtained by Doob (1953) [5] 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 twoparameter 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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198608406243
PII:
S 00029939(1986)08406243
Keywords:
HardyLittlewood maximal function,
,
,
martingale inequalities,
Doob
Article copyright:
© Copyright 1986 American Mathematical Society
