We study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [I. BialynickiBirula, Formulation of the uncertainty relations in terms of the RΓ©nyi entropies, Physical Review A 74 (5) (2006) 052101] and Zozor et al. [S. Zozor, C. Vignat, On classes of nonGaussian asymptotic minimizers in entropic uncertainty principles, Physica A 375 (2) (2007) 499-517]. Those inequalities can be considered as generalizations of the Heisenberg uncertainty principle, since they measure the mutual uncertainty of a wave function and its Fourier transform through their associated RΓ©nyi entropies with conjugated indices. We consider here the general case where the entropic indices are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indices (\alpha) and (\beta) in the plane ((\alpha, \beta)). Our results explain and extend a recent study by Luis [A. Luis, Quantum properties of exponential states, Physical Review A 75 (2007) 052115], where states with quantum fluctuations below the Gaussian case are discussed at the single point ((2,2)).