Hypertranscendency of Perturbations of Hypertranscendental Functions

Inspired by the work of Bank on the hypertranscendence of $Gamma e^h$ where $Gamma$ is the Euler gamma function and $h$ is an entire function, we investigate when a meromorphic function $fe^g$ cannot satisfy any algebraic differential equation over certain field of meromorphic functions, where $f$ and $g$ are meromorphic and entire on the complex plane, respectively. Our results (Theorem 1 and 2) give partial solutions to Banks Conjecture (1977) on the hypertranscendence of $Gamma e^h$. We also give some sufficient conditions for hypertranscendence of meromorphic function of the form $f+g$, $fcdot g$ and $fcirc g$ in Theorem 3 and 4.