Is there a positive bound from below, independent of $\lambda$, on the expression
$\frac{1+|\lambda|^2}{2}|z| + \Re(\lambda z)$
where $z$ is a nonzero complex number, and $\lambda$ is a complex number such that $|\lambda|<1$?
Here the script $\Re$ is meant to mean the real part of $\lambda z$. I wasn't sure how to type it in the standard notation.
No.
$|\operatorname{re} (\lambda z ) | \le |\lambda | |z|$, hence ${1 \over 2} (1+ |\lambda|^2) |z| + \operatorname{re} (\lambda z ) \ge {1 \over 2} (1+ |\lambda|^2) |z| - |\lambda | |z| = {1 \over 2} (1-|\lambda|)^2 |z|$.
To see why no bound exists, take $z=1$ and $\lambda \in (0,1)$. Then ${1 \over 2} (1+ |\lambda|^2) |z| + \operatorname{re} (\lambda z ) = {1 \over 2} (1-\lambda)^2$, and $\inf_{\lambda \in (0,1)} {1 \over 2} (1-\lambda)^2 = 0$.