Upper bound of the best constant of the Trudinger-Moser inequality and its application to the Gagliardo-Nirenberg inequality

We will consider a Trudinger-Moser inequality for the critical Sobolev space H n/p,p (R n ) with the fractional derivatives in R n and obtain an upper bound of the best constant of such an inequality. Moreover, by changing normalization from the homogeneous norm to the inhomogeneous one, we will give the best constant in the Hilbert space H n/2,2 (R n ). As an application, we will obtain some lower bound of the best constant of a Gagliardo-Nirenberg inequality.