Abstract
Examining the asymptotic growth behavior of holomorphic and meromorphic functions has significant importance in complex analysis. Estimations of upper bounds for the rate of growth of entire functions are mostly guaranteed. However, the analog estimations for lower bounds of the rate of growth are not always attainable. In this paper, we give the lower rate of growth of the generalized Hadamard product of two entire axially monogenic functions. It has also been shown that the product entire axially monogenic function is of regular growth or perfectly regular growth when its constituent entire axially monogenic functions possess these properties. The investigation of both upper and lower bounds by means of linear transmutation is also provided. Furthermore, some applications related to approximation theory are outlined.