Relative (p,q)−φ order based some growth analysis of composite p-adic entire functions

Abstract

Let $\mathbb{K}$ be a complete ultrametric algebraically closed field and $\mathcal{A}\left(\mathbb{K}\right)$\ be the $\mathbb{K}$-algebra of entire function 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 $sup\{|f\left(x\right)|:|x|=r\}$ where $|\cdot |(r)$ is a multiplicative norm on $\mathcal{A}\left(\mathbb{K}\right).$ In this paper we study some growth properties of composite $p$-adic entire functions on the basis of their relative $(p,q)$-$\phi$ order where $p$, $q$ are any two positive integers and $\phi \left(r\right)$ $:$ $[0,+\mathrm{\infty })\to (0,+\mathrm{\infty })$ is a non-decreasing unbounded function of $r$.