$(p,q)$th order oriented growth measurement of composite $p$-adic entire functions

Abstract

Let $\mathbb{K}$ be a complete ultrametric algebraically closed field and let $\mathcal{A}\left(\mathbb{K}\right)$ be the $\mathbb{K}$-algebra of entire functions on $\mathbb{K}$. For any $p$-adic entire function $f\in \mathcal{A}\left( \mathbb{K}\right)$ and $r>0$, we denote by $|f|\left(r\right)$ the number $\sup \left\{ |f\left( x\right) |:|x|=r\right\}$, where $\left\vert \cdot \right\vert (r)$ is a multiplicative norm on $\mathcal{A}\left( \mathbb{K}\right)$. For any two entire functions $f\in \mathcal{A}\left(\mathbb{K}\right)$ and $g\in \mathcal{A}\left(\mathbb{K}\right)$ the ratio $\frac{|f|(r)}{|g|(r)}$ as $r\rightarrow \infty$ is called the comparative growth of $f$ with respect to $g$ in terms of their multiplicative norms. Likewise to complex analysis, in this paper we define the concept of $(p,q)$th order (respectively $(p,q)$th lower order) of growth as $\rho ^{\left( p,q\right) }\left( f\right) =\underset{r\rightarrow +\infty }{\lim \sup } \frac{\log ^{[p]}|f|\left( r\right) }{\log ^{\left[ q\right] }r}$ (respectively $\lambda ^{\left( p,q\right) }\left( f\right) =\underset{ r\rightarrow +\infty }{\lim \inf }\frac{\log ^{[p]}|f|\left( r\right) }{\log ^{\left[ q\right] }r}$), where $p$ and $q$ are any two positive integers. We study some growth properties of composite $p$-adic entire functions on the basis of their $\left(p,q\right)$th order and $(p,q)$th lower order.