Problem 1518 (difficulty: 5/10)

Let \(\displaystyle p(z)\in \C[z]\) be of degree at least 1. Prove the following

(a) If all roots of \(\displaystyle p\) have negative real parts, then \(\displaystyle \re\dfrac{p'(z)}{p(z)} >0\).

(b) If the roots of \(\displaystyle p(z)\) all lie in the half plane \(\displaystyle \re z<0\), then the same holds for \(\displaystyle p'(z)\).

(c) (Gauss) If \(\displaystyle p(z)\in \C[z]\), then the roots of \(\displaystyle p'\) are contained in the convex hull of the roots of \(\displaystyle p\).

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