  Number-phase uncertainty relations are formulated in terms of unified
entropies which form a family of two-parametric extensions of the Shannon
entropy. For two generalized measurements, unified-entropy uncertainty
relations are given in both the state-dependent and state-independent forms. A
few examples are discussed as well. Using the Pegg--Barnett formalism and the
Riesz theorem, we obtain a nontrivial inequality between norm-like functionals
of generated probability distributions in finite dimensions. The principal
point is that we take the infinite-dimensional limit right for this inequality.
Hence number-phase uncertainty relations with finite phase resolutions are
expressed in terms of the unified entropies. Especially important case of
multiphoton coherent states is separately considered. We also give some
entropic bounds in which the corresponding integrals of probability density
functions are involved.
