Some results on zeros of a polynomial

Résumé

For a given polynomial
$p\left( z\right) =a_{0}+a_{1}z+\cdots +a_{n}z^{n}$
with real or complex coefficients, Gulzar $\left( 2015\right) $ proved
for any real $p>1,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, all the
zeros of $p\left( z\right) $ lie in the closed circular region%
\begin{equation*}
\left\vert z\right\vert \leq \max \left\{ L_{p},L_{p}^{\frac{1}{n+1}%
}\right\},
\end{equation*}%
where%
\begin{equation*}
L_{p}=\left( n+1\right) ^{\frac{1}{q}}\left( \sum\limits_{j=1}^{n}\left\vert
\frac{a_{n-1}a_{n-j}-a_{n}a_{n-j-1}}{a_{n}^{2}}\right\vert ^{p}\right) ^{%
\frac{1}{p}},\text{ }a_{-1}=0.
\end{equation*}%
Recently, Gulzar also generalized the above result by using a lacunary type polynomial.
Unfortunately, the proof is not correct. In this present paper an attempt has been taken to investigate
and extend the above results were made. Moreover, we have produce some important
corollaries which give the generalization of Mezerji and Bidkham $\left( 2014\right)$ results.