Zeros of Lacunary Type Polynomials

Abstract

Using Schwarz's lemma, Mohammad (1965) proved that all zeros of the polynomial \(f(z) = a0 + a1z + · · · + an−1z^{n−1} + anz^n\) with real or complex coefficients lie in the closed disc \( |z| ≤ M' / |an|\) if \( |an| ≤ M'\), where \( M' = max|z|=1|a0 + a1z + · · · + an−1z^{n−1}|\). In this paper, we present new results on the location of zeros of the lacunary type polynomial \(p(z) = a0 + a1z + · · · + apz^p + anz^n, p < n\). In particular, for \( p = n−1\), our first result implies an important corollary which sharpens the above result. Also, we described some regions in which all zeros of \(p(z)\) are simple. In many cases, our results give better bounds for the location of polynomial zeros than the known ones.