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 $\delta (p,M)$.