Reciprocal polynomials with all zeros on the unit circle (Q653844)

Let \(f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_0\in\mathbb R[x]\) be a reciprocal polynomial of degree \(d\). It is proved that if the coefficient vector \((a_d,a_{d-1},\dots,a_0)\) (or \((a_{d-1},\dots,a_1)\)) is close enough in the \(l_1^{d+1}\) (or \(l_1^{d-1}\)) distance to the constant vector \((b,b,\dots,b)\in {\mathbb R}^{d+1}\) (or \((b,b,\dots,b)\in {\mathbb R}^{d-1}\)) then all zeros of \(f(x)\) are on the unit circle. Two examples show that this sufficient condition is applicable even if the condition given by \textit{P. Lakatos} and \textit{L. Losonczi} [Math. Inequal. Appl. 10, No. 4, 761--769 (2007; Zbl 1130.30005)] is not.