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A complex polynomial

Source: Indian Postal Coaching 2005

September 23, 2005
algebrapolynomialalgebra unsolved

Problem Statement

Let f(z)=amzn+an−1zn−1+⋯+a1z+a0f(z) = a_m z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0 be a polynomial of degree n≥3n \geq 3 with real coefficients.Suppose all roots of f(z)=0f(z) =0 lie in the half plane  z∈C:Re(z)<0}{\ z \in \mathbb{C} : Re(z) < 0 \}}. Prove that akak+3<ak+1ak+2a_k a_{k+3} < a_{k+1}a_{k+2} for k=0,1,2,3,....n−3k = 0,1,2,3,.... n-3
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