## 6 Observations and Discussion Regarding MW  
Let us now discuss the findings of our study on the applied side. First, consider \(q_{4}(z)=z^{6}+\frac{1}{4}z^{5}+\frac{1}{8}z^{4}+\frac{1}{16}z^{3}+\frac{1}{2 5}z^{2}+\frac{1}{36}z+\frac{1}{49}\); we find that the largest zero has modulus \(=\mathbf{0.5447544053}\). Table 4 shows that our result **MW** is pretty close to the exact modulus and it is much better than all other upper bounds. Another example shows the efficiency of our result (31), consider \(q_{5}(z)=z^{6}+\frac{1}{3}z^{4}+\frac{1}{4}z^{3}+\frac{1}{9}z^{2}+\frac{1}{100}\); we find that the largest zero has modulus \(=\mathbf{0.7419983061}\). Table 5 shows that our result **MW** is very close to the exact modulus. After careful considerations and investigations, for almost all cases of \(c_{k}\)'s (with \(|c_{k}|<1\)), we discovered that (31) is very effective as long as \(|c_{k}|<1\), besides the other mentioned assumptions. The following example explains why we chose the presented conditions in (31), and thus it solidifies and supports the reason for our selection. Let  
\[h_{1}(z)=z^{6}+\frac{1}{6}z^{4}+\frac{1}{5}z^{2}+\frac{1}{4}.\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Mathematician** & **Upper Bound** \\ \hline [PERSON] (6) & 1.25 \\ [PERSON] and [PERSON] (7) & 1.039971167 \\ [PERSON] (8) & 1 \\ [PERSON] and [PERSON] (9) & 1.066738881 \\ [PERSON] (10) & 1.198213950 \\ [PERSON] (11) & 2.091031073 \\ [PERSON] (12) & 1.152835774 \\ [PERSON] and [PERSON] (13) & 1.072449189 \\ [PERSON] et al. (14) & 1.573586825 \\ Theorem 4 & 1.219108946 \\
**MW** & \(\mathbf{0.6721175730}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparisons of upper bounds based on the considered polynomial \(q_{4}(z)\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Mathematician** & **Upper Bound** \\ \hline [PERSON] (6) & 1.333333333 \\ [PERSON] and [PERSON] (7) & 1.089062344 \\ [PERSON] (8) & 1 \\ [PERSON] and [PERSON] (9) & 0.9939972629 \\ [PERSON] (10) & 1.167303296 \\ [PERSON] (11) & 2.078873251 \\ [PERSON] (12) & 1.325435041 \\ [PERSON] and [PERSON] (13) & 1.299097566 \\ [PERSON] et al. (14) & 1.581696908 \\ Theorem 4 & 1.351454829 \\
**MW** & \(\mathbf{0.7647166222}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparisons of upper bounds based on the considered polynomial \(q_{5}(z)\). The real coefficients of \(h_{1}(z)\) do not respect our conditions in (31). So, the largest zero has modulus \(=0.8120242973\), but the upper bound in (31) \(=0.7685824855\), which is an incorrect upper bound, implying that (31) does not apply arbitrary for any \(c_{k}\)'s unless we have some restriction(s). To ensure the correctness--and after long extrapolation by testing many cases--we established the investigated assumptions in (31) for all polynomials with real coefficients. In this regard, it is worth noting that the assumptions about \(c_{k}\)'s, whenever \(|c_{k}|<1\), are only valid for real coefficients. However, if some of \(c_{k}\)'s is complex, then it is not true that we cannot apply (31). For example, the polynomial  
\[h_{2}(z)=z^{6}+\bigg{(}\frac{1}{4}+i\frac{1}{4}\bigg{)}z^{5}+\frac{i}{9}z^{4}+ \frac{i}{16}z^{3}+\frac{1}{25}z^{2}+\frac{1}{36}z+\frac{1}{49},\]  
refuting three assumptions of (31). Namely, we have that some \(c_{k}\)'s is complex, with \(\sum\limits_{k=2}^{n}|c_{k}|=0.5949422794\leq\frac{2}{3}\), and \(|c_{3}|<|c_{2}|\). At the same time, we find that the largest zero has a modulus \(=0.6408240287\). However, the upper bound in (31) \(=0.7337440145\), which means that (31) still can be applied under other conditions as long as \(|c_{k}|<1\). We are not able to determine the sufficient condition for complex coefficients in the case that \(|c_{k}|<1\). Furthermore, the polynomial  
\[h_{3}(z)=z^{6}+\frac{1}{4}z^{4}+\frac{1}{3}z^{2}+\frac{1}{4},\]  
is a good example, showing that the conditions \(|c_{k+1}|<|c_{k}|\) and \(\sum\limits_{k=2}^{n}|c_{k}|=0.5833333333\leq\frac{2}{3}\) are not necessary but they are sufficient.