Operations preserving coefficient majorisation

Write $f=\sum a_{\alpha }{z}^{\alpha }$, $g=\sum c_{\alpha }{z}^{\alpha }$, $F=\sum A_{\alpha }{z}^{\alpha }$, $G=\sum C_{\alpha }{z}^{\alpha }$. For every $\alpha$, $\mathrm{\mid }{a}_{\alpha }+c_{\alpha }\mathrm{\mid }\le A_{\alpha }+C_{\alpha }$ and $\mathrm{\mid }{\sum }_{\beta +\gamma =\alpha }{a}_{\beta }{c}_{\gamma }\mathrm{\mid }\le {\sum }_{\beta +\gamma =\alpha }{A}_{\beta }{C}_{\gamma }$. Each convolution is finite, including degree zero, proving the sum and product claims.

The coefficient of $z^{\alpha }$ in ${\mathrm{\partial }}_{j}f$ is $({\alpha }_j+1)a_{\alpha +e_j}$; its modulus is at most $({\alpha }_j+1)A_{\alpha +e_j}$, the coefficient in ${\mathrm{\partial }}_{j}F$. Thus differentiation preserves the relation.

In degree at most $q$, a product $h^{\beta }$ can contribute only when $\mathrm{\mid }\beta \mathrm{\mid }\le q$, since every factor has order at least one. There are finitely many such $\beta$ and finitely many decompositions of any fixed multi-index. Repeated application of step 1.1 bounds each coefficient of $a_{\beta }{h}^{\beta }$ by the corresponding coefficient of $A_{\beta }{H}^{\beta }$. Adding these finitely many bounds proves the composition claim.

For convergent inner series choose a positive smaller polyradius $s$ for which the sums $S_i(s)={\sum }_{\alpha }\mathrm{\mid }[z^{\alpha }]h_i\mathrm{\mid }{s}^{\alpha }$ are strictly below the outer convergence radii. Such an $s$ exists because $S_i(s)\to 0$ as all coordinates of $s$ decrease to zero. The absolute sum of the expanded substitution is bounded by ${\sum }_{\beta }\mathrm{\mid }{a}_{\beta }\mathrm{\mid }{\prod }_i{S}_i(s{)}^{{\beta }_i}<\mathrm{\infty }$. Absolute rearrangement therefore identifies the formal substitution with the actual function composition; the same argument applies to products. On still smaller polydiscs F2 identifies the formal derivatives with derivatives of the sums.

Write $f=a+h$ with $a\mathrm{\ne }0$ and $h(0)=0$. Shrink $s$ until ${\sum }_{\alpha }\mathrm{\mid }[z^{\alpha }](h\mathrm{/}a)\mathrm{\mid }{s}^{\alpha }<1$. F3 bounds the absolute sum of $a^{-1}{\sum }_{k\ge 0}(-h\mathrm{/}a{)}^k$ by $\mathrm{\mid }a{\mathrm{\mid }}^{-1}\mathrm{/}(1-S)$ with $S<1$. The finite geometric identity gives $(a+h)a^{-1}{\sum }_{k=0}^K(-h\mathrm{/}a{)}^k=1-(-h\mathrm{/}a{)}^{K+1}$; the last term tends uniformly to zero on this polydisc. Hence the convergent series is exactly $1\mathrm{/}f$. This proves the reciprocal and all asserted analytic-operation claims.