Zeros of lacunary type polynomials

Using Schwarz's lemma, Mohammad (1965) proved that all zeros of the polynomial
$$ f(z)=a_0+a_1z+\dots+a_{n-1}z^{n-1}+a_nz^n $$
with real or complex coefficients lie in the closed disc
$$ |z|\leqslant\frac{M'}{|a_n|}\text{ if } |a_n|\leqslant M', $$
where
$$ M'=\max_{|z|=1}|a_0+a_1z+\dots+a_{n-1}z^{n-1}|. $$
In this paper, we present new results on the location of zeros of the lacunary type polynomial
$$ p(z)=a_0+a_1z+\dots+a_pz^p+a_nz^n,\quad p<n. $$
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.