  In this paper we derive the best constant for the following
Gagliardo-Nirenberg interpolation inequality
  \begin{eqnarray*} \|u\|_{L^{m+1}}\leq C_{q,m,p}
\|u\|^{1-\theta}_{L^{q+1}}\|\nabla u\|^{\theta}_{L^p},\quad
\theta=\frac{pd(m-q)}{(m+1)[d(p-q-1)+p(q+1)]}, \end{eqnarray*} where parameters
$q,m,p$ respectively belong to the following two ranges:
  (i) $p>d\geq 1$, $q\geq0$ and $m=\infty$. That shows $L^{\infty}$-type
Gagliardo-Nirenberg interpolation inequality.
  (ii) $p>\max\{1,\frac{2d}{d+2}\}$, $0\leq q<\sigma-1$, and $q<m<\sigma$,
where $\sigma$ is defined by $ \sigma:= \frac{(p-1)d+p }{d-p}$ if $p<d$;
$\sigma:=\infty $ if $p\geq d$. That gives $L^{m}$-type Gagliardo-Nirenberg
interpolation inequality.
  The best constant $C_{q,m,p}$ is given by \begin{eqnarray*}
C_{q,m,p}:=\theta^{-\frac{\theta}{p}}(1-\theta)^{\frac{\theta}{p}-\frac{1}{m+1}}M_c^{-\frac{\theta}{d}},\quad
M_c:=\int_{\mathbb{R}^d}u_{c,m}^{q+1}\,dx, \end{eqnarray*} where $u_{c,m}$ is
the unique radial non-increasing solution to a generalized Lane-Emden equation.
The case of equality holds when $u=Au_{c,m}(\lambda(x-x_0))$ for any real
numbers $A>0$, $\lambda >0$ and $x_{0}\in \mathbb{R}^d$.
  In particular, for the case $m=+\infty$, the generalized Lane-Emden equation
becomes a Thomas-Fermi type equation. For $q=0,~m=\infty$ or $d=1$, $u_{c,m}$
are closed form solutions expressed in term of the incomplete Beta functions.
Moreover, we show that $u_{c,m}\to u_{c,\infty}$ and $C_{q,m,p}\to
C_{q,\infty,p}$ as $m\to +\infty$ for $d=1$.
