Let be the unit ball in , be the unit disc in . It is well known, that for any holomorphic function the following Schwarz inequality holds
There are many generalizations of this result. One of them is the following one due to S. Y. Dai and Y. F. Pan [Proc. Am. Math. Soc. 136, No. 2, 635–640 (2008; Zbl 1133.30005)], which gives the estimate of higher order derivatives: if is holomorphic, then
The authors generalize this result on several complex variables as follows. Let be holomorphic, . Then for any multiindex
where , .