Question

Let us define $\mathbb{C}_+:=\{z\in \mathbb{C}:\Re{z}>0\}$, where $\mathbb{C}$ is the field of complex number, and $\Re{z}$ represents the real part of the complex number $z$. Let $a,b$ be two real numbers such that $b>b_0*a>a_0$ where $a_0>0$ and $b_0>1$. There exists a positive constant $C$, independent on a, b, and z, (but it can depend on $a_0$ ) such that \begin{equation} |\frac{z^3+z^2+bz+a}{z+1}|\geq C a \qquad \text{for every $z\in \mathbb{C}_+$ and for every $b>b_0*a>a_0$} \end{equation} where $| \cdot|$ represents the modulus of a complex number????????????? Notice that, it has been already shown that there exists a positive constant $K=K(a_0)$, independent on a, b, and z, such that \begin{equation} |\frac{z^3+z^2+bz+a}{z+1}|\geq K \sqrt{a} \qquad \text{for every $z\in \mathbb{C}_+$ and for every $b>b_0*a>a_0$} \end{equation}

0