On the Interpolation of Analytic Mappings

Let $(E_0,E_1)$ and $(H_0,H_1)$ be two pairs of complex Banach spaces densely and continuously embedded into each other, $E_1\hookrightarrow E_0$ and $H_1\hookrightarrow H_0$ and also let $\|x\|_{E_0} \le \|x\|_{E_1}$. By $E_\theta=[E_0,E_1]_\theta$ and $H_\theta=[H_0,H_1]_\theta$ we denote the spaces obtained by the complex interpolation method for $\theta\in[0,1]$, and by $B_\theta (0,R)$ we denote an open ball of radius $R$ in the space $E_\theta$. Let $\Phi\colon B_0(0,R)\to H_0$ be an analytic mapping taking $B_1(0,R)$ into $H_1$, and let the estimates
$$ \|\Phi(x)\|_{H_\theta} \le C_\theta\|x\|_{H_\theta}\qquad \text{for all}\quad x\in B_\theta(0,R) $$
hold for $\theta = 0,\,1$. Then, for all $\theta\in (0,1)$, the mapping $\Phi$ takes the ball $B_\theta(0,r)$ of radius $r\in(0,R)$ in the space $E_\theta$ into $H_\theta$ and
$$ \|\Phi(x)\|_{H_\theta}\le C_0^{1-\theta}C_1^\theta \frac{R}{R-r}\|x\|_{E_\theta}, \qquad x\in B_\theta(0,r). $$