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Entire functions and some of their growth properties on the basis of generalized order (α,β) | ||
| International Journal of Nonlinear Analysis and Applications | ||
| دوره 12، شماره 2، بهمن 2021، صفحه 1735-1747 اصل مقاله (430.13 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2021.22299.2346 | ||
| نویسندگان | ||
| Tanmay Biswas1؛ Chinmay Biswas2؛ Biswajit Saha* 3 | ||
| 1Rajbari, Rabindrapally, R. N. Tagore Road P.O. Krishnagar, Dist-Nadia, PIN- 741101, West Bengal, India | ||
| 2Department of Mathematics,Nabadwip Vidyasagar College, Nabadwip, Dist.- Nadia, PIN-741302, West Bengal, India | ||
| 3Department of Mathematics, Government General Degree College Muragachha, Nakashipara, Dist.- Nadia, PIN-741154,West Bengal, India | ||
| تاریخ دریافت: 14 دی 1399، تاریخ بازنگری: 28 بهمن 1399، تاریخ پذیرش: 14 اسفند 1399 | ||
| چکیده | ||
| For any two entire functions $f$, $g$ defined on finite complex plane $\mathbb{C}$, the ratios $\frac{M_{f\circ g}(r)}{M_{f}(r)}$ and $\frac{M_{f\circ g}(r)}{M_{g}(r)}$ as $r\rightarrow \infty $ are called the growth of composite entire function $f\circ g$ with respect to $f$ and $g$ respectively in terms of their maximum moduli. Several authors have worked about growth properties of functions in different directions. In this paper, we have discussed about the comparative growth properties of $f\circ g$, $f$ and $g,$ and derived some results relating to the generalized order $(\alpha ,\beta )$ after revised the original definition introduced by Sheremeta, where $\alpha ,$ $\beta $ are slowly increasing continuous functions defined on $(-\infty,+\infty )$. Under different conditions, we have found the limiting values of the ratios formed from the left and right factors on the basis of their generalized order $(\alpha ,\beta )$ and generalized lower order $(\alpha,\beta ),$ and also established some inequalities in this regard. | ||
| کلیدواژه‌ها | ||
| Entire function؛ growth؛ composition؛ generalized order $(\alpha, \beta ) $؛ generalized lower order $(\alpha,\beta )$ | ||
| مراجع | ||
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[1] T. Biswas and C. Biswas, Generalized order (α, β) oriented some growth properties of composite entire functions, Ural Math. J. 6(2) (2020) 25–37. [2] T. Biswas and C. Biswas, Some results on generalized relative order (α, β) and generalized relative type (α, β) of meromorphic function with respective to an entire function, Ganita, 70(2) (2020) 239–252. [3] T. Biswas and C. Biswas and R. Biswas, A note on generalized growth analysis of composite entire functions, Poincare J. Anal. Appl. 7(2) (2020) 277–286. [4] R. Biswas, Some growth properties based on (p,q)-th order of the integer translation of the composite entire and meromorphic functions, Korean J. Math. 26(4) (2018) 561–582. [5] D. Banerjee and B. Mandal, Generalised iteration of entire functions with index-pair [p,q], Kanuralp J. Math. 5(1) (2017) 132–144. [6] D. Banerjee and N. Mondal, On the growth of iterated entire functions, Int. J. Anal. Appl. 4(1) (2014) 21–25. [7] I. Chyzhykov and N. Semochko, Fast growing entire solutions of linear differential equations, Math. Bull. Shevchenko Sci. Soc. 13 (2016) 68–83. [8] J. Clunie, The Composition of Entire and Meromorphic Functions, Mathematical Essays dedicated to A. J. Macintyre, Ohio University Press, 1970. [9] 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. [10] B.Y. Levin, Y. Lyubarskii, M. Sodin and V. Tkachenko, Lectures on entire functions, Translations of Mathematical Monographs, American Mathematical Society, 1996. [11] L. A. Rubel, Entire and meromorphic functions, Springer-Verlag, 1996. [12] D. Sato, On the rate of growth of entire functions of fast, growth, Bull. Am. Math. Soc. 69 (1963) 411–414. [13] M.N. Sheremeta, On the growth of a composition of entire functions, Carpathian Math. Publ. 9(2) (2017) 181–187. [14] A.P. Singh, On maximum term of composition of entire functions, Proc. Nat. Acad. Sci. India, 59(A) Part I(1989) 103–115. [15] A. P. Singh and M. S. Baloria, On the maximum modulus and maximum term of composition of entire functions, Indian J. Pure Appl. Math. 22(12) (1991) 1019–1026. [16] M.N. Sheremeta, Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion, Izv. Vyssh. Uchebn. Zaved Mat. 2 (1967) 100–108. [17] G. Valiron, Lectures on The General Theory of Integral Functions, Chelsea Publishing Company, New York, 1949. [18] H. Y. Xu, J. Tu and C. F. Yi, The applications on some inequalities of the composition of entire functions, J. Math. Inequal. 7(4) (2013) 759–772. [19] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993. | ||
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