en.wikipedia.org/wiki/Entropic_uncertainty

For the Fourier transform normalized as (\hat f(\xi)=\int_{\mathbb R} e^{-2\pi i x\xi}f(x),dx), the sharp Hausdorff–Young/Babenko–Beckner operator norm in one dimension is

[
|\mathcal F|_{q,p}=\left(\frac{p^{1/p}}{q^{1/q}}\right)^{1/2}, \qquad 1<p\le 2,\ \frac1p+\frac1q=1.
]

So the flat text here omits the square root and states a larger constant than the true one. A standard reference summary of Beckner’s sharp inequality gives the constant in dimension (n) as ((p^{1/p}(p')^{-1/p'})^{n/2}); in dimension 1 with (p'=q), that is exactly ((p^{1/p}/q^{1/q})^{1/2}).

The one-dimensional maximum-entropy bound for a density with variance (V(\phi)) is

[
H(\phi) \le \frac12\log\bigl(2\pi e,V(\phi)\bigr)
]

with equality for a Gaussian. The flat text here drops the factor (1/2) (or, equivalently, the square root inside the logarithm), changing the inequality.

Why this matters: for a Gaussian with variance (\sigma^2), the entropy is exactly (\tfrac12\log(2\pi e\sigma^2)). So the correct Gaussian-equality case is incompatible with the printed formula unless that missing factor is restored.

Combining the entropic uncertainty bound with the Gaussian maximum-entropy bound gives

[
\frac12 \le \frac{\exp(H(|f|^2)+H(|g|^2))}{2\pi e} \le \sqrt{V(|f|^2)V(|g|^2)}.
]

So the flat text is missing the square root on the right.

This is not just a cosmetic typo: the printed version would imply the stronger lower bound (V(|f|^2)V(|g|^2)\ge 1/2), which is false. For minimum-uncertainty Gaussians, equality holds in the correct inequality and one has (\sqrt{V(|f|^2)V(|g|^2)}=1/2), i.e. (V(|f|^2)V(|g|^2)=1/4), not (\ge 1/2).