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VJIMC 2019 P4 Category II

Source: VJIMC 2019

March 30, 2019
complex analysisreal analysiscomplex numbersVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic

Problem Statement

Let D={z∈C:Im⁡(z)>0,Re⁡(z)>0}D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \} . Let n≥1n \geq 1 and let a1,a2,…an∈Da_1,a_2,\dots a_n \in D be distinct complex numbers. Define f(z)=z⋅∏j=1nz−ajz−aj‾f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}} Prove that f′f' has at least one root in DD.
Proposed by Géza Kós (Lorand Eotvos University, Budapest)
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