Bounded multivariable power series are holomorphic with unique derivative coefficients

For $0<s<1$ and $\mathrm{\mid }{z}_i-a_i\mathrm{\mid }\le s{r}_i$, the coefficient bound gives $\mathrm{\mid }{c}_{\alpha }(z-a{)}^{\alpha }\mathrm{\mid }\le M{s}^{\mathrm{\mid }\alpha \mathrm{\mid }}$. The sum of these bounds over the box ${\alpha }_i\le N$ is $M{\prod }_{i<m}{\sum }_{j=0}^N{s}^j\le M(1-s{)}^{-m}$. Every finite set of multi-indices lies in a box, so [F2] makes the bounds summable, and [F1] and the M-test give the asserted absolute uniform convergence. Each point of ${\mathrm{\Delta }}_r(a)$ lies in such a closed polydisc, so $g$ is defined throughout ${\mathrm{\Delta }}_r(a)$.

Fix $z$ with $w=z-a$ and $\mathrm{\mid }{w}_i\mathrm{\mid }\le s{r}_i$, where $0<s<t<1$. For $h\mathrm{\ne }0$ put $\eta ={\max }_{i<m}\mathrm{\mid }{h}_i\mathrm{\mid }\mathrm{/}{r}_i$ and restrict to $h$ small enough that $T=(t-s)\mathrm{/}\eta \ge 1$. This is possible since $\eta \le \parallel h\parallel \mathrm{/}{\min }_i{r}_i$. Expand the polynomial ${\varphi }_{\alpha }(u)={\prod }_{i<m}(w_i+u{h}_i{)}^{{\alpha }_i}={\sum }_j{A}_{\alpha ,j}{u}^j$ by [F4]. Its first two coefficients are $A_{\alpha ,0}=w^{\alpha }$ and $A_{\alpha ,1}={\sum }_{k<m,{\alpha }_k>0}{\alpha }_k{w}^{\alpha -e_k}{h}_k$. The triangle inequality and the nonnegative binomial coefficients give ${\sum }_j\mathrm{\mid }{A}_{\alpha ,j}\mathrm{\mid }{T}^j\le {\prod }_{i<m}(\mathrm{\mid }{w}_i\mathrm{\mid }+T\mathrm{\mid }{h}_i\mathrm{\mid }{)}^{{\alpha }_i}\le {\prod }_{i<m}(t{r}_i{)}^{{\alpha }_i}$. As $T\ge 1$, the remainder $R_{\alpha }={\varphi }_{\alpha }(1)-A_{\alpha ,0}-A_{\alpha ,1}$ therefore satisfies $\mathrm{\mid }{R}_{\alpha }\mathrm{\mid }\le T^{-2}{\prod }_{i<m}(t{r}_i{)}^{{\alpha }_i}$.

Fix $k<m$ and $0<\theta <1$. Reindexing by $\alpha =\nu +e_k$ gives $\mathrm{\mid }{d}_{\nu }\mathrm{\mid }\le (M\mathrm{/}{r}_k)({\nu }_k+1){\prod }_{i<m}{r}_i^{-{\nu }_i}=(M\mathrm{/}{r}_k)({\nu }_k+1){\theta }^{\mathrm{\mid }\nu \mathrm{\mid }}{\prod }_{i<m}(\theta r_i{)}^{-{\nu }_i}$. For a natural $j$, $(j+1){\theta }^j={\sum }_{p+q=j}{\theta }^{p+q}\le ({\sum }_{p=0}^j{\theta }^p{)}^2\le (1-\theta {)}^{-2}$. Since ${\theta }^{\mathrm{\mid }\nu \mathrm{\mid }}\le {\theta }^{{\nu }_k}$, this proves the claimed bound on $d_{\nu }$. Step 1.1 applied to $d$ on radii $\theta r$ gives absolute uniform convergence of its series on every closed $sr$ polydisc with $s<\theta$. For each $s<1$ such a $\theta$ exists; thus the first derived series converges absolutely and uniformly on every smaller closed polydisc.

Step 1.2 and the box estimate in step 1.1 imply ${\sum }_{\alpha }\mathrm{\mid }{c}_{\alpha }{R}_{\alpha }\mathrm{\mid }\le M{\eta }^2\mathrm{/}((t-s{)}^2(1-t{)}^m)\le M\parallel h{\parallel }^2\mathrm{/}(({\min }_i{r}_i{)}^2(t-s{)}^2(1-t{)}^m)$. Both $z$ and $z+h$ belong to the closed $tr$ polydisc, and the first derived sums $b_k(z)={\sum }_{\nu }({\nu }_k+1)c_{\nu +e_k}{w}^{\nu }$ converge absolutely by step 2.1. Thus [F1] permits splitting and reindexing to obtain $g(z+h)-g(z)-{\sum }_{k<m}{b}_k(z)h_k={\sum }_{\alpha }{c}_{\alpha }{R}_{\alpha }$. The map $h\mapsto {\sum }_k{b}_k(z)h_k$ is complex-linear, and the displayed bound is $O(\parallel h{\parallel }^2)$, hence $o(\parallel h\parallel )$. By [F3], $g$ has this total complex differential at $z$. Taking $h=u{e}_k$ and $h=iu{e}_k$ with real $u\to 0$ gives ${\mathrm{\partial }}_{x_k}g=b_k$ and ${\mathrm{\partial }}_{y_k}g=i{b}_k$, so ${\mathrm{\partial }}_{z_k}g=b_k$. Every interior $z$ admits such $s,t$, proving holomorphy and the first termwise derivative formula throughout ${\mathrm{\Delta }}_r(a)$.

The arguments in steps 1.1, 1.2, 2.1 and 3.1 used only the stated coefficient bound, so they apply again to any derived coefficient family with a bound on positive radii. Fix a multi-index $\gamma$ of order $q=\mathrm{\mid }\gamma \mathrm{\mid }>0$ and a real $0<\tau <1$. Put $t_j=1-j(1-\tau )\mathrm{/}q$ for $0\le j\le q$, so $1=t_0>t_1>\cdots >t_q=\tau >0$. Apply step 2.1 successively with radii $t_{j-1}r$ and shrink factor $t_j\mathrm{/}{t}_{j-1}$, and apply step 3.1 at each stage. The finitely many constants in the bounds remain finite. Induction on the number of differentiations gives the coefficient at $\nu$ after differentiating with multi-index $\delta$ as $((\nu +\delta )!\mathrm{/}\nu !)c_{\nu +\delta }$: one further differentiation in coordinate $k$ multiplies the coefficient at $\nu +e_k$ by ${\nu }_k+1$, changing the factorial denominator from $(\nu +e_k)!$ to $\nu !$. This formula is independent of the order of coordinate differentiations. The final series represents ${\mathrm{\partial }}_z^{\gamma }g$ and is holomorphic on ${\mathrm{\Delta }}_{\tau r}(a)$ by step 3.1. For any closed $sr$ polydisc with $s<1$, choose $s<\tau <1$; step 1.1 applied to the final coefficient bound gives absolute uniform convergence there. Since every interior point lies in some ${\mathrm{\Delta }}_{\tau r}(a)$, all these identities and holomorphy hold throughout ${\mathrm{\Delta }}_r(a)$. For $q=0$ they are step 1.1 and step 3.1.

At $z=a$ every monomial in the derived series of step 4.1 vanishes except $\nu =0$, whose value is $1$; thus ${\mathrm{\partial }}_z^{\gamma }g(a)=\gamma !c_{\gamma }$. If $c^{\mathrm{\prime }}$ satisfies the stated bound and has the same sum, the argument just proved applies to $c^{\mathrm{\prime }}$ as well. The derivatives are those of the same function, so $\gamma !c_{\gamma }=\gamma !c_{\gamma }^{\mathrm{\prime }}$. Cancelling the nonzero factorial gives $c_{\gamma }=c_{\gamma }^{\mathrm{\prime }}$ for every $\gamma$, proving uniqueness.