  Classical interpolation inequality of the type $\|u\|_{X}\leq
C\|u\|_{Y}^{\theta}\|u\|_{Z}^{1-\theta}$ is well known in the case when $X$,
$Y$, $Z$ are Lebesgue spaces. In this paper we show that this result may be
extended by replacing norms $\|\cdot\|_{Y}$ or $\|\cdot\|_{X}$ by suitable H\"
older semi-norm. We shall even prove sharper version involving weak Lorentz
norm. We apply this result to prove the Gagliardo--Nirenberg inequality for a
wider scale of parameters.
