$n\ge2$ is a given positive integer,and $z_{1},z_{2},\cdots,z_{n}\in \mathbb{C}$,such $$|z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2\ge n$$

prove or disprove $$f_{n}=\sum_{j=1}^{n}\left|\sum_{I\subseteq \{1,2,3,\cdots,n\},|I|=j}\prod_{k\in I}z_{k}\right|^2\ge 1$$

In the particular case when $n=2$can be proved \begin{align*}f_{2}&=|z_{1}+z_{2}|^2+|z_{1}z_{2}|^2=(|z_{1}|^2+|z_{2}|^2)+|z_{1}z_{2}|^2+2\Re(z_{1}\overline{z_{2}})\\ &\ge |z_{1}|^2+|z_{2}|^2+|z_{1}z_{2}|^2-2|z_{1}z_{2}|\\ &=(|z_{1}|^2+|z_{2}|^2-1)+(|z_{1}z_{2}|-1)^2\\ &\ge(|z_{1}|^2+|z_{2}|^2-1)\\ &\ge 1 \end{align*}

Let $n\ge2$ be a given positive integer, and $z_{1},z_{2},\cdots,z_{n}\in \mathbb{C}$,such $$|z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2\ge n.$$ Prove or disprove $$f_{n}=\sum_{j=1}^{n}\left|\sum_{I\subseteq \{1,2,3,\cdots,n\},|I|=j}\prod_{k\in I}z_{k}\right|^2\ge 1$$

In the particular case when $n=2$, it can be proved that \begin{align*}f_{2}&=|z_{1}+z_{2}|^2+|z_{1}z_{2}|^2=(|z_{1}|^2+|z_{2}|^2)+|z_{1}z_{2}|^2+2\Re(z_{1}\overline{z_{2}})\\ &\ge |z_{1}|^2+|z_{2}|^2+|z_{1}z_{2}|^2-2|z_{1}z_{2}|\\ &=(|z_{1}|^2+|z_{2}|^2-1)+(|z_{1}z_{2}|-1)^2\\ &\ge(|z_{1}|^2+|z_{2}|^2-1)\\ &\ge 1 \end{align*}

$n\ge3$ is a given positive integer,and $z_{1},z_{2},\cdots,z_{n}\in \mathbb{C}$,such $$|z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2\ge n$$

prove or disprove $$\sum_{j=1}^{n}\left|\sum_{I\subseteq \{1,2,3,\cdots,n\},|I|=j}\prod_{k\in I}z_{k}\right|^2\ge 1$$

$n\ge2$ is a given positive integer,and $z_{1},z_{2},\cdots,z_{n}\in \mathbb{C}$,such $$|z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2\ge n$$

prove or disprove $$f_{n}=\sum_{j=1}^{n}\left|\sum_{I\subseteq \{1,2,3,\cdots,n\},|I|=j}\prod_{k\in I}z_{k}\right|^2\ge 1$$

In the particular case when $n=2$can be proved \begin{align*}f_{2}&=|z_{1}+z_{2}|^2+|z_{1}z_{2}|^2=(|z_{1}|^2+|z_{2}|^2)+|z_{1}z_{2}|^2+2\Re(z_{1}\overline{z_{2}})\\ &\ge |z_{1}|^2+|z_{2}|^2+|z_{1}z_{2}|^2-2|z_{1}z_{2}|\\ &=(|z_{1}|^2+|z_{2}|^2-1)+(|z_{1}z_{2}|-1)^2\\ &\ge(|z_{1}|^2+|z_{2}|^2-1)\\ &\ge 1 \end{align*}