Formulation of the uncertainty relations in terms of the Rényi entropies

Quantum-mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the Rényi entropies. The proof of these inequalities requires the use of the exact expression for the $(p,q)$-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in $N$-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized Rényi entropies.