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1 $begingroup$ $f_1 , f_2 , ... , f_n $ is a sequence of holomorphic function in an open set $Omega$ , and also $$|f_1|+...+|f_n|$$ attains its maximum in $Omega$ . Can we prove that each of $f_k$ is constant ? My attempt : If $n=1$ , we can find a $theta$ such that $f_1' =f_1 e^{i theta}$ attains its maximum in $Omega$ , then $f_1 ' $ is a constant so $f_1$ is a constant . If $n gt 1$ , then we can find a sequence $theta_1 ,..., theta_n$ such that $f_1 e^{i theta_1} +...+ f_n e^{i theta_n}$ attains its maximum in $Omega$ . Let $f_k'=e^{i theta_k} f_k$ , we find that $$|f_1'|+...+|f_k'| =|f_1|+...+|f_n|$$ So , to prove both $f_k$ are constant , it suffice to prove that following statement : $g_1 , ... , g_n$ is a sequence of holomorphic function in an open set $Omega$