The following generalization of Rosenthal's inequality was proved by Burkholder: A⁻¹ₚ\\|s(f)\|ₚ + \|d^\|ₚ\ ≤ \|f^\|ₚ ≤ Bₚ\\|s(f)\|ₚ + \|d^\|ₚ\, for all martingales (fₙ). It is known that Aₚ grows like √p as p → ∞. In this paper we prove that the growth rate of Bₚ as p → ∞ is p/ln p.
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Paweł Hitczenko (1990) studied this question.
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