 Solution

The printed assertion is false. Take real $w=1-\epsilon$ with $0<\epsilon <1$. The principal complex logarithm is unambiguous, butNo finite $C_3$ can satisfy the proposed estimate, even on these points within the logarithm's domain. At $w=1$ the logarithm is undefined, and the usual principal analytic branch also excludes its cut. These are additional reasons the literal all-$w$ assertion cannot hold.

The valid estimate for the final product proof is insteador equivalently $|1-w|\le e^{|w|}$ for every $w$, including $w=1$. The upper bound on the real part of the logarithm is enough for growth; a bound on the full modulus of the logarithm is unnecessary. The preceding root Solution supplies the complete convergence and exponential-growth proof using this correction.