Let $B$ be the unit ball in $\mathbb{R}^n$ and define $$ \widetilde{B} = \{\lambda x : x\in B, \lambda \in \mathbb{C}, \lvert \lambda \rvert \leq 1\}. $$ Now, suppose that $u$ is holomorphic in some neighbourhood of $\widetilde{B}$ such that $\sup\limits_{\widetilde{B}} \lvert u(z)\rvert \leq 1$.
For the case $n=1$, I was able to show (for any $N\in\mathbb{N}$) that: $$ \left\lvert u(z) - \sum_0^{2N-1}\frac{u^{(k)}(0)z^k}{k!} \right\rvert \leq (2N+1)\lvert{z}\rvert^{2N} $$ and $$ \left\lvert \frac{u^{(k)}(0)z^k}{k!} \right\rvert \leq \lvert{z}\rvert^{k}. $$
According to the notes I am reading, "apply the preceding inequalities to all complex lines passing through $0$, and we obtain for all $z\in\widetilde{B}$": $$ \left\lvert u(z) - \sum_{\lvert\alpha\rvert \leq 2N-1}\frac{u^{(\alpha)}(0)z^\alpha}{\alpha!} \right\rvert \leq (2N+1)\lvert{z}\rvert^{2N} $$ and $$ \left\lvert\sum_{\lvert\alpha\rvert=k}\frac{u^{(\alpha)}(0)z^\alpha}{\alpha!}\right\rvert \leq \lvert{z}\rvert^{k}.$$
I do not see how these inequalities are obtained. I am relatively new to multi-dimensional holomorphic functions and would appreciate if anyone could shed some light on how to do this.
Thanks in advance!