Relative (p,q)-φ order and relative (p,q)-φ type based on some growth properties of composite p-adic entire functions

Biswas, Tanmay

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creative_2020_29_1_09_16

DOI: https://doi.org/10.37193/CMI.2020.01.02

Issue no: Vol 29/2020 no. 1 Tags: Growth, $p$-adic entire function, composition, relative $\left( p, q\right) $-$\varphi $ order, q\right) $% -$\varphi $ type, q\right) $-$\varphi $ weak type, index-pair

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Let $\mathbb{K}$ be a complete ultrametric algebraically closed field and $\mathcal{A}\left( \mathbb{K}\right)$ 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 $\mathcal{A}\left( \mathbb{K}\right)$ . In this paper we study some growth properties of composite p-adic entire functions on the basis of their relative $\left( p,q\right)$ – $\varphi$ order, relative $\left( p,q\right)$ – $\varphi$ type and relative $\left(p,q\right)$ – $\varphi$ weak type where $p, q$ are any two positive integers and $\varphi \left( r\right)$ $:$ $[0,+\infty)\rightarrow (0,+\infty )$ is a non-decreasing unbounded function of $r$ .