Let (E0,E1) and (H0,H1) be two pairs of complex Banach spaces densely and continuously embedded into each other, E1 ↪ E0 and H1 ↪ H0 and also let \(\left\| x \right\|_{E_0 } \leqslant \left\| x \right\|_{E_1 } \). By Eθ = [E0, E1]θ and Hθ = [H0, H1]θ we denote the spaces obtained by the complex interpolation method for θ ∈ [0, 1], and by Bθ(0,R) we denote an open ball of radius R in the space Eθ. Let Φ: B0(0,R) → H0 be an analytic mapping taking B1(0,R) into H1, and let the estimates $$\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_\theta \left\| x \right\|_{H_\theta } for allx \in B_\theta (0,R)$$ hold for θ = 0, 1. Then, for all θ ∈ (0, 1), the mapping Φ takes the ball Bθ(0,r) of radius r ∈ (0,R) in the space Eθ into Hθ and $$\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_0^{1 - \theta } C_1^\theta \frac{R} {{R - r}}\left\| x \right\|_{E_\theta } ,x \in B_\theta (0,r). $$ .