Polynomial estimates and radius of analyticity on real Banach spaces

A generalization of Problem 73 of Mazur and Orlicz in the Scottish Book was introduced from L. Harris. The exact value of the constant that appears there is known when complex normed linear spaces are considered. In this paper, we give estimates in the case of an arbitrary real normed linear space and a real ${\mathrm{\ell }}_p$ space. Moreover, if $F(x)$ is a power series, $\rho$ its radius of uniform convergence and ${\rho }_{\begin{array}{c}A\end{array}}$ its radius of analyticity, we prove that ${\rho }_{\begin{array}{c}A\end{array}}\ge \rho \mathrm{/}\sqrt{2}$ and give some respective results for the $n$th Fr'{e}chet derivative of $F(x)$.