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

T. Biswas

Abstract


Let us consider $\mathbb{K}$ be a complete ultrametric algebraically closed field and suppose $\mathcal{A}\left( \mathbb{K}\right) $ be the $\mathbb{K}$-algebra of entire functions on $\mathbb{K}$. For any $p$-adic entire functions $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 another $p$-adic entire functions $g\in \mathcal{A}\left( \mathbb{K}\right),$ $|g|\left(r\right) $ is defined and the ratio $\frac{|f|\left( r\right) }{|g|\left( r\right) }$ as $r\rightarrow \infty $ is called the comparative growth of $f$ with respect to $g$ in terms of their multiplicative norm. 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 then 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 where $p$ and $q$ are any two positive integers.

Keywords


p-adic entire function, growth, $(p,q)$th order, $(p,q)$th lower order, composition

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