|g(re^{i\varphi})|\leq|1+b_kz^k|+(|b_{k+1}z^{k+1}|+\cdots+|b_nz^n|)=\\
=|1-r^k|b_k||+r^{k+1}(|b_{k+1}|+\cdots+|b_n|r^{n-k-1})=\\
=1-r^k(|b_k|-r|b_{k+1}|-\cdots-r^{n-k}|b_n|)<1