  In this paper we provide a new set of uncertainty principles for unitary
operators using a sequence of inequalities with the help of the
geometric-arithmetic mean inequality. As these inequalities are "fine-grained"
compared with the well-known Cauchy-Schwarz inequality, our framework naturally
improves the results based on the latter. As such, the unitary uncertainty
relations based on our method outperform the best known bound introduced in
[Phys. Rev. Lett. 120, 230402 (2018)] to some extent. Explicit examples of
unitary uncertainty relations are provided to back our claims.
