This entry can be considered as a continued part to a recent entry where lots of significantly important inequalities (Hardy, Opial, Rellich, Serrin, Caffarelli–Kohn–Nirenberg, Gagliardo-Nirenberg-Sobolev, Horgan) have been considered.

Today we shall continue to list here more important inequalities in the literature. Given , , , define the number by

where

.

**Gagliardo-Nirenberg’s inequality**. For any one has

.

When and , Gagliardo-Nirenberg’s inequality then becomes the well-known Sobolev inequality.

**Sobolev’s inequality**. For any one has

.

The best constant has been obtained by Aubin [here] and Talenti [here], independently. Namely, they showed that

and

where is the volume of the unit ball in and the gamma function.

When , , and , Gagliardo-Nirenberg’s inequality then becomes the well-known Nash inequality.

**Nash’s inequality**. For any one has

.

The best constant for the Nash inequality is given by

where is the first non-zero Neumann eigenvalue of the Laplacian operator in the unit ball. This come from a joint work between Carlen and Loss [here].

Another consequence of Gagliardo-Nirenberg’s inequality is the logarithmic Sobolev inequality.

**logarithmic Sobolev’s inequality**. For any one has

where also satisfies

.

In fact, it can be obtained as the limit case when , that is, and , . To see this, let us first notice the fact that the constant in Gagliardo-Nirenberg’s inequality is independent of . We can rewrite Gagliardo-Nirenberg’s inequality as

where . It then follows that

.

Thus when we get

where we have used the fact that the function

satisfies

.

Therefore, replacing and writing , we obtain the logarithmic Sobolev’s inequality.

The best constant for the logarithmic Sobolev inequality is given by

.

We refer the reader to a book due to Hebey entitled “*Nonlinear analysis on manifolds: Sobolev spaces and inequalities*” for details.

The best constant for the Gagliardo-Nirenberg inequality is not completely solved. In some cases, we was able to find its best constants [here, here].

See also: Sobolev type inequalities on Riemannian manifolds