Ordering among the topologies induced by various polynomial norms

Abstract

Let us consider the following two norms in the vector space \,$\mathcal{P}$ \,of all complex polynomials: $$ \|p\|_{D_{r}}:= \sup\{|p(z)|: |z|<r\}, \text{ and } \|p\|_{1}:=\sum_{i=0}^{n}|a_{i}|, $$ where \,$p(z) = \sum_{i=0}^{n} a_{i} z^{i}$. In this note we show that, if $0 <\varepsilon < \varepsilon' < 1 < r < r'$, then $$ \|\cdot\|_{D_{\varepsilon}}\prec\|\cdot\|_{D_{\varepsilon'}}\prec\|\cdot\|_{D_{1}}\prec \|\cdot\|_{1}\prec \|\cdot\|_{D_{r}}\prec \|\cdot\|_{D_{r'}}, $$ where \,$\prec$ \,represents the natural (strict) partial order in their corresponding induced topologies.