The Minimal Length and the Shannon Entropic Uncertainty Relation

In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relation [X,P]=iħ(1+βP²), where β is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycielski (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, that is, X=x and P=tan (√βp)/√β, where [x,p]=iħ, the BBM inequality is still valid in the form S_x+S_p≥1+ln π as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.