On zero of entire functions with majorant of infinite order

The class of entire functions $f$ satisfying $\ln|f(z)|\le C_f\lambda (|z|)$, $z\in \mathbb{C}$, where $\lambda$ is a majorant mentioned in the title, admits a zero set $\{z_j\}$ such that $\{|z_j|\}$ is a uniqueness set for this class.