Meromorphic Functions Sharing a Nonzero Value with their Derivatives

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 1,  2015, pp.137-147
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.1.137
Title & Authors
Meromorphic Functions Sharing a Nonzero Value with their Derivatives
Li, Xiao-Min; Ullah, Rahman; Piao, Da-Xiong; Yi, Hong-Xun;

Abstract
Let f be a transcendental meromorphic function of finite order in the plane such that $\small{f^{(m)}}$ has finitely many zeros for some positive integer $\small{m{\geq}2}$. Suppose that $\small{f^{(k)}}$ and f share a CM, where $\small{k{\geq}1}$ is a positive integer, $\small{a{\neq}0}$ is a finite complex value. Then f is an entire function such that $f^{(k)}-a Keywords Meromorphic functions;Order of growth;Shared values;Uniqueness theorems; Language English Cited by References 1. A. Al-Khaladi, On meromorphic functions that share one value with their derivative, Analysis, 25(2005), 131-140. 2. A. Al-Khaladi, On meromorphic functions that share one small function with their kth derivative, Results. Math., 57(2010), 313-318. 3. A. Al-Khaladi, Meromorphic functions that share one finite value CM or IM with their k-th derivative, Results. Math., 63(2013), 95-105. 4. A. Banerjee and P. Bhattacharjee, Uniqueness of meromorphic functions sharing one value with their derivatives, Mathematical Communications, 13(2008), 277-288 . 5. R. Bruck, On entire functions which share one value CM with their first derivative, Results in Math., 30(1996), 21-24. 6. J. M. Chang and Y. Z. Zhu, Entire functions that share a small function with their derivatives, J. Math. Anal. Appl., 351(2009), 491-496. 7. J. M. Chang and M. L. Fang, Entire functions that share a small function with their derivatives, Complex Variables Theory Appl., 49(2004), 871-895. 8. Z. X. Chen and K. H. Shon, On conjecture of R. Bruck concerning the entire function sharing one value CM with its derivative, Taiwanese J. Math., 8(2004), 235-244. 9. Z. X. Chen and C. C. Yang, Some further results on the zeros and growths of entire solutions of second order linear differential equation, Kodai Math J., 22(1999), 273-285. 10. Z. X. Chen, The growth of solutions of$f^{{\prime}{\prime}}+e^{-2}f^{\prime}+Q(z)f=0$, Science in China (A), 31(2001), 775-784. 11. G. G. Gundersen and L. Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl., 223(1998 ), 88-95. 12. W. K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964. 13. J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and J. Zhang, Value sharing results for shifts of meromorphic functions and sufficient conditions for periodicity, J. Math. Anal. Appl., 355(2009), 352-363. 14. G. Jank and L. Volkmann, Einfuhrung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhauser, Basel-Boston, 1985. 15. I. Lahiri and A. Sarkar, Uniqueness of a meromorphic function and its derivative, J. Inequal. Pure and Appl. Math., 5(2004), Issue 1, Art. 20. 16. I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin/New York, 1993. 17. J. K. Langley, The second derivative of a meromorphic function of finite order, Bulletin London Math. Soc., 35(2003), 97-108. 18. J. K. Langley, Proof of a conjecture of Hayman concerning f and$f^{{\prime}{\prime}}\$, Bulletin London Math. Soc., 48(1993), 500-514.

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