On the zeros of lacunary-type polynomials

Abstract

Let $$p\ge 2$$ be an integer, $$M>0$$ be a real number and $$\begin{aligned} {\mathcal {C}}(p,M)= & {} \Bigl \{ z^n + a_{n-p} z^{n-p} + \cdots + a_1 z +a_0 \, \Big | \,\\&\max _{0\le j\le n-p} |a_j| =M, \, n=p, p+1, \ldots \Bigr \}, \end{aligned}$$ where the coefficients $$a_j$$ $$(j= 0, 1,\ldots ,n-p)$$ are complex numbers. Guggenheimer (Am Math Mon 71:54–55, 1964) and Aziz and Zargar (Proc Indian Acad Sci 106:127–132, 1996) proved that if $$P\in {\mathcal {C}}(p,M)$$ , then all zeros of P lie in the disk $$|z|<\delta (p,M)$$ , where $$\delta (p,M)$$ is the only positive solution of $$x^p-x^{p-1}=M$$ . We show that $$\delta (p,M)$$ is the best possible value. Moreover, we present some monotonicity/concavity/convexity properties and limit relations of $$\delta (p,M)$$ .