Let (X0​,X1​) and (Y0​,Y1​) be complex Banach couples and assume that
X1​⊆X0​ with norms satisfying ∥x∥X0​​≤c∥x∥X1​​ for
some c>0. For any 0<θ<1, denote by Xθ​=[X0​,X1​]θ​
and Yθ​=[Y0​,Y1​]θ​ the complex interpolation spaces and by
B(r,Xθ​), 0≤θ≤1, the open ball of radius r>0 in
Xθ​, centered at zero. Then for any analytic map Φ:B(r,X0​)→Y0​+Y1​ such that Φ:B(r,X0​)→Y0​ and Φ:B(c−1r,X1​)→Y1​
are continuous and bounded by constants M0​ and M1​, respectively, the
restriction of Φ to B(c−θr,Xθ​), 0<θ<1, is
shown to be a map with values in Yθ​ which is analytic and bounded by
M01−θ​M1θ​