Majorant series ($f\ll g$)

A power series $g(x)={\sum }_{\alpha }{b}_{\alpha }{x}^{\alpha }$ with nonnegative coefficients majorizes $f(x)={\sum }_{\alpha }{a}_{\alpha }{x}^{\alpha }$ when $|a_{\alpha }|\le b_{\alpha }$ for every multi-index $\alpha$. Coefficientwise majorization turns estimates for an analytic equation into estimates for a simpler positive-coefficient equation.