  We derive strong variance-based uncertainty relations for arbitrary two and
more unitary operators by re-examining the mathematical foundation of the
uncertainty relation. This is achieved by strengthening the celebrated
Cauchy-Schwarz inequality using a method of brackets and convex functions. The
unitary uncertainty relations outperform several strong unitary uncertainty
relations, notably better than some recent best lower bounds such as [Phys.
Rev. Lett. 120, 230402 (2018)] and [Phys. Rev. A. 100, 022116 (2019)].
