On zeros of polynomial

For a given polynomial
\begin{equation*} P\left( z\right) =z^{n}+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\cdots +a_{1}z+a_{0} \end{equation*}
with real or complex coefficients, the Cauchy bound
\begin{equation*} \left\vert z\right\vert <1+A,\qquad A=\underset{0\leqslant j\leqslant n-1}{ \max }\left\vert a_{j}\right\vert \end{equation*}
does not reflect the fact that for $A$ tending to zero, all the zeros of $P\left( z\right) $ approach the origin $z=0$. Moreover, Guggenheimer (1964) generalized the Cauchy bound by using a lacunary type polynomial
\begin{equation*} p\left( z\right) =z^{n}+a_{n-p}z^{n-p}+a_{n-p-1}z^{n-p-1}+\cdots +a_{1}z+a_{0}, \qquad 0<p<n\text{.} \end{equation*}
In this paper we obtain new results related with above facts. Our first result is the best possible. For the case as $A$ tends to zero, it reflects the fact that all the zeros of $P(z)$ approach the origin $z=0$; it also sharpens the result obtained by Guggenheimer. The rest of the related results concern zero-free bounds giving some important corollaries. In many cases the new bounds are much better than other well-known bounds.