The aim of the article is to establish a high order Schwarz lemma for holomorphic mappings between the unit balls and in complex Hilbert spaces and , respectively. For and , we denote by the quantity
For a holomorphic mapping , we denote by the Fréchet derivative of order at a point and by its evaluation at a -tuple of vectors in .
Then the authors’ first main result is as follows:
Theorem 1: Let be a holomorphic mapping. Then, for any positive integer , any and , one has
With the same method of proof as for Theorem 1, the authors obtain
Theorem 2: For holomorphic mappings with a positive real part, one has
for , an integer , and any .
Also, the norm of the -th order Fréchet derivative is estimated in the following two theorems, namely:
Theorem 3: Let be a holomorphic mapping. Then for any positive integer and any , one has
Finally, they obtain for functions as in Theorem 2:
Theorem 4: For holomorphic mappings with a positive real part one has, with integer valued , for the estimate