On the zeros of lacunary-type polynomials

Let $$p\ge 2$$ p ≥ 2 be an integer, $$M>0$$ M > 0 be a real number and $$\begin{aligned} {\mathcal {C}}(p,M)= & {} \Bigl \{ z^n + a_{n-p} z^{n-p} + \cdots + a_1 z +a_0 \, \Big | \,\\&\max _{0\le j\le n-p} |a_j| =M, \, n=p, p+1, \ldots \Bigr \}, \end{aligned}$$ C ( p , M ) = { z n + a n - p z n - p + ⋯ + a 1 z + a 0 | max 0 ≤ j ≤ n - p | a j | = M , n = p , p + 1 , … } , where the coefficients $$a_j$$ a j $$(j= 0, 1,\ldots ,n-p)$$ ( j = 0 , 1 , … , 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 $$P\in {\mathcal {C}}(p,M)$$ P ∈ C ( p , M ) , then all zeros of P lie in the disk $$|z|<\delta (p,M)$$ | z | < δ ( p , M ) , where $$\delta (p,M)$$ δ ( p , M ) is the only positive solution of $$x^p-x^{p-1}=M$$ x p - x p - 1 = M . We show that $$\delta (p,M)$$ δ ( p , M ) is the best possible value. Moreover, we present some monotonicity/concavity/convexity properties and limit relations of $$\delta (p,M)$$ δ ( p , M ) .