We obtain an improvement of the Beckner inequality valid for and the Gaussian measure. Our improvement is essential for the intermediate case , and moreover, we find the natural extension of the inequality for any real .
"Improving Beckner's bound via Hermite functions." Anal. PDE 10 (4) 929 - 942, 2017. https://doi.org/10.2140/apde.2017.10.929