May be the picture can not be unfold, this question below is in page11 of book Fourier analysis and and nonlinear partial differential analysis Bahouri-Chemin-Danchin.
When we study harmonic analysis, Riesz-Thorin interpolation theorem is very important. The proof of Riese-Thorin complex interpolation is easy because we just consider simple functions, and we can easily conclude that $F(z)$ in the following picture is analytic because $a_{k}^{P(z)}b_{j}^{Q(z)}$ is analytic($a_{k}$ and $b_{j}$ are constant).
But, in the above picture, how can we prove that $F(z)$ is holomophic and bounded(note that in the above picture, the author does not use simple functions)?
Any hints are wellcome thanks everyone who read this question!!!