The minimal length and the Shannon entropic uncertainty relation

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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 Mycieslki (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, i.e., X=x and P=(√βp)/√β where [x,p]=iℏ, the BBM inequality is still valid in the form Sₓ+Sₚ≥1+π as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.