  Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and
Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases.
First we establish interpolation inequalities under quite general assumptions
on the parameters of the function spaces. Several results on necessary
conditions are also provided. Next, utilizing the interpolation inequalities
together with some embedding results, we prove Gagliardo-Nirenberg inequalities
for fractional derivatives in Lorentz spaces, which do hold even for the
limiting case when one of the parameters is equal to 1 or $\infty$.
