Problem 1520 (difficulty: 7/10)

Let \(\displaystyle n\ge2\) and \(\displaystyle u_1=1,u_2,\ldots,u_n\) be complex numbers with absolute value at most \(\displaystyle 1\), and let

\(\displaystyle f(z)=(z-u_1)(z-u_2)\ldots(z-u_n). \)

Show that the polynomial \(\displaystyle f'(z)\) has a root with nonnegative real part.

KöMaL A. 430.

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