## Beckner type of the logarithmic Sobolev and a new type of Shannon’s inequalities and an application to the uncertainty principle

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- by Hideo Kubo, Takayoshi Ogawa and Takeshi Suguro
- Proc. Amer. Math. Soc.
**147**(2019), 1511-1518 - DOI: https://doi.org/10.1090/proc/14350
- Published electronically: December 19, 2018
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## Abstract:

We consider the inequality which has a “dual” relation with Beckner’s logarithmic Sobolev inequality. By using the relative entropy, we identify the sharp constant and the extremal of this inequality. Moreover, we derive the logarithmic uncertainty principle like Beckner’s one.## References

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## Bibliographic Information

**Hideo Kubo**- Affiliation: Faculty of Science, Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
- MR Author ID: 345544
**Takayoshi Ogawa**- Affiliation: Mathematical Institute/Research Alliance Center of Mathematical Science, Tohoku University, Sendai 980-8578, Japan
- MR Author ID: 289654
**Takeshi Suguro**- Affiliation: Mathematical Institue, Tohoku University, Sendai 980-8578, Japan
- Received by editor(s): May 31, 2018
- Published electronically: December 19, 2018
- Additional Notes: The work of the first author was partially supported by JSPS Grant-in-aid for Scientific Research S #16H06339.

The work of the second author was partially supported by JSPS Grant-in-aid for Scientific Research S #25220702 and Challenging Research (Pioneering) #17H06199. - Communicated by: Joachim Krieger
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 1511-1518 - MSC (2010): Primary 26D10, 39B62
- DOI: https://doi.org/10.1090/proc/14350
- MathSciNet review: 3910416