A brief look at the statement of Gagliardo-Nirenberg interpolation inequality would suggest that there should exist a proof by a clever use of Jensen's inequality. In other words, there should be a function of the form $$F(a-b/n)=\|D^b f\|_a$$ such that $\log F(a-b/n)$ is convex. That would provide a short proof of the inequality. The existing proofs that I could find on the internet involved complicated double inductions, etc. Is anyone aware of a proof along these lines? Thanks!

Fourier Analysis and Nonilinear Partial Differential Equationsby H. Bahouri, J.-Y. Chemin and R. Danchin (Springer-Verlag, 2011). $\endgroup$