Littlewood's 4/3 inequality

In mathematical analysis, Littlewood's 4/3 inequality, named after John Edensor Littlewood,[1] is an inequality that holds for every complex-valued bilinear form defined on , the Banach space of scalar sequences that converge to zero.

Precisely, let or be a bilinear form. Then the following holds:

where

The exponent 4/3 is optimal, i.e., cannot be improved by a smaller exponent.[2] It is also known that for real scalars the aforementioned constant is sharp.[3]

Generalizations

Bohnenblust–Hille inequality

Bohnenblust–Hille inequality[4] is a multilinear extension of Littlewood's inequality that states that for all -linear mapping the following holds:

See also

References
  1. Littlewood, J. E. (1930). "On bounded bilinear forms in an infinite number of variables". The Quarterly Journal of Mathematics. os-1 (1): 164–174. Bibcode:1930QJMat...1..164L. doi:10.1093/qmath/os-1.1.164.
  2. Littlewood, J. E. (1930). "On bounded bilinear forms in an infinite number of variables". The Quarterly Journal of Mathematics (1): 164–174. Bibcode:1930QJMat...1..164L. doi:10.1093/qmath/os-1.1.164.
  3. Diniz, D. E.; Munoz, G.; Pellegrino, D.; Seoane, J. (2014). "Lower bounds for the Bohnenblust--Hille inequalities: the case of real scalars". Proceedings of the American Mathematical Society (132): 575–580. arXiv:1111.3253. doi:10.1090/S0002-9939-2013-11791-0. S2CID 119128323.
  4. Bohnenblust, H. F.; Hille, Einar (1931). "On the Absolute Convergence of Dirichlet Series". The Annals of Mathematics. 32 (3): 600–622. doi:10.2307/1968255. JSTOR 1968255.


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