Multiplicative comparisons of Rényi entropies for weighted Bernoulli sums

Abstract: We establish improved multiplicative bounds relating the Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic bound between the zeroth-order and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain, Sah, and Sawhney. Additionally, we obtain explicit constant-factor bounds for comparisons among Rényi entropies of nonzero orders.