Relative $(p,q)-\varphi$ order based some growth analysis of composite $p$-adic entire functions
Main Article Content
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 \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 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 $r$.
Article Details
References
[1] L. Bernal, Orden relativo de crecimiento de funciones enteras, Collect. Math. 39 (1988), 209– 229. Google Scholar
[2] Google Scholar
[3] T. Biswas, Relative (p, q)-φ order oriented some growth properties of p-adic entire functions, J. Fract. Calc. Appl., 11 (1) (2020), 161–169. Google Scholar
[4] Google Scholar
[5] T. Biswas, Some growth properties of composite p-adic entire functions on the basis of their relative order and relative lower order, Asian-Eur. J. Math., 12 (3) (2019), 1950044, 15p., https://doi.org/10.1142/S179355711950044X. Google Scholar
[6] Google Scholar
[7] T. Biswas, Some growth aspects of composite p-adicentire functions in the light of their (p, q)-th relative order and (p, q)-th relative type, J. Chungcheong Math. Soc., 31 (4) (2018), 429–460. Google Scholar
[8] Google Scholar
[9] T. Biswas, On some growth analysis of p-adic entire functions on the basis of their (p,q)-th relative order and (p, q)-th relative lower order, Uzbek Math. J., 2018 (4) (2018), 160–169. Google Scholar
[10] Google Scholar
[11] T. Biswas, Relative order and relative type based growth properties of iterated p-adic entire functions, Korean J. Math., 26 (4) (2018), 629–663. Google Scholar
[12] Google Scholar
[13] T. Biswas, A note on (p, q)-th relative order and (p, q)-th relative type of p-adic entire functions, Honam Math. J., 40 (4) (2018), 621–659. Google Scholar
[14] Google Scholar
[15] T. Biswas,(p,q)-th order oriented growth measurement of composite p-adic entire functions, Carpathian Math. Publ., 10 (2) (2018), 248–272. Google Scholar
[16] Google Scholar
[17] K. Boussaf, A. Boutabaa and A. Escassut, Order, type and cotype of growth for p-Adic entire functions, A survey with additional properties, p-Adic Numbers, Ultrametric Anal. Appl.,8 (4), (2016), 280–297. Google Scholar
[18] Google Scholar
[19] K. Boussaf, A. Boutabaa and A. Escassut, Growth of p-adic entire functions and applications, Houston J. Math., 40 (3) (2014), 715–736. Google Scholar
[20] Google Scholar
[21] A. Escassut, K. Boussaf and A. Boutabaa, Order, type and cotype of growth for p-adic entire functions, Sarajevo J. Math., Dedicated to the memory of Professor Marc Krasner, 12(25) (2) (2016), 429–446, suppl. Google Scholar
[22] Google Scholar
[23] A. Escassut, Value Distribution in p-adic Analysis, World Scientific Publishing Co. Pte. Ltd., Singapore, 2015. Google Scholar
[24] Google Scholar
[25] A. Escassut and J. Ojeda, Exceptional values of p-adic analytic functions and derivative, Complex Var. Elliptic Equ., 56 (1-4) (2011), 263–269. Google Scholar
[26] Google Scholar
[27] A. Escassut,p-adic Value Distribution. Some Topics on Value Distribution and Differentability in Complex and P-adic Analysis, Math. Monogr., Series 11, Science Press, Beijing, (2008), 42–138. Google Scholar
[28] Google Scholar
[29] A. Escassut, Analytic Elements in p-adic Analysis, World Scientific Publishing Co. Pte. Ltd. Singapore, 1995. Google Scholar
[30] Google Scholar
[31] P. C. Hu and C. C. Yang, Meromorphic Functions over non-Archimedean Fields, Kluwer Aca- demic Publishers, (publ. city), 2000. Google Scholar
[32] Google Scholar
[33] O. P. Juneja, G. P. Kapoor and S. K. Bajpai, On the (p,q)-order and lower (p,q)-order of an entire function, J. Reine Angew. Math., 282 (1976), 53–67. Google Scholar
[34] Google Scholar
[35] A. Robert, A Course in p-adic analysis, Graduate texts, Springer (publ. city), 2000. Google Scholar
[36] Google Scholar
[37] X. Shen, J. Tu and H. Y. Xu, Complex oscillation of a second-order linear differential equa- tion with entire coefficients of [p,q]-φ order, Adv. Difference Equ, 2014:200 (2014), 14 p., http://www.advancesindifferenceequations.com/content/2014/1/200. Google Scholar
[38] Google Scholar