Coefficientwise majorisation

Fix a centre $c$, $n\ge 1$ coordinates, and ordinary coefficients as in Multi-indexed power series in ${\mathbb{C}}^m$ and their absolute convergence. Put $w=z-c$. For formal expansions $f(c+w)={\sum }_{\alpha }{a}_{\alpha }{w}^{\alpha }$ and $G(c+w)={\sum }_{\alpha }{b}_{\alpha }{w}^{\alpha }$, write $f{\ll }_{c}G$ if for every $\alpha \in {\mathbb{N}}^n$ we have $\mathrm{\mid }{a}_{\alpha }\mathrm{\mid }\le b_{\alpha }$. When the centre is clear, write simply $f\ll G$. In particular every $b_{\alpha }$ is real and nonnegative. For analytic series $a_{\alpha }=D^{\alpha }f(c)\mathrm{/}\alpha !$; these are not the unnormalized derivatives. At $c=0$ this is the usual expansion in $z$. Vectors are compared component by component and must have the same number of components. No convergence is required for this relation.

Gantumur, §2 equation (12), printed p. 5; §3 equation (28), printed p. 7.