Does the logarithm inequality extend to the complex plane?

For estimates, the inequality $\mathrm{log}(y)\le y-1,$$y>0$ is often helpful. Is there any sort of upper bound for the logarithm function in the complex plane? Specifically, $|\mathrm{log}(z)|\le $ something for all $z\in \mathbb{C}$

Perhaps this would work?: $\mathrm{log}(z)\le \sqrt{{\mathrm{log}}^{2}|z|+\mathrm{arg}(z{)}^{2}}$

For estimates, the inequality $\mathrm{log}(y)\le y-1,$$y>0$ is often helpful. Is there any sort of upper bound for the logarithm function in the complex plane? Specifically, $|\mathrm{log}(z)|\le $ something for all $z\in \mathbb{C}$

Perhaps this would work?: $\mathrm{log}(z)\le \sqrt{{\mathrm{log}}^{2}|z|+\mathrm{arg}(z{)}^{2}}$