On the upper bounds for the constants of the Hardy–Littlewood inequality

The best known upper estimates for the constants of the Hardy–Littlewood inequality for m  -linear forms on ℓpℓp spaces are of the form (2)m−1. We present better estimates which depend on p and m  . An interesting consequence is that if p≥m2p≥m2 then the constants have a subpolynomial growth as m tends to infinity.