On the transcendental meromorphic solutions of a certain class of
differential equations

S. Majumder; A. Dam

doi:10.15330/ms.51.1.130-142

In this paper we consider the differential equation \[F^{(k)}-a_{2}=e^{\gamma}\{\alpha(F-a_{1})+\beta\},\] where $a_{i}(z), \alpha(z)(\not\equiv 0,\infty)$, $\beta(z)(\not\equiv \infty)$, $i=1,2$ are small functions of $F$, $\gamma$ is an entire function and $k\in\mathbb{N}$. Let $f$ be a transcendental meromorphic function such that $ N(r,\infty;f)=S(r,f)$ and $n\in\mathbb{N}$ such that $n\geq k+1$. If $F=f^{n}$ is a solution of the above differential equation, then \[F^{(k)}\equiv \frac{a_{2}\alpha}{a_{1}\alpha-\beta}F.\] Also we exhibit an example to fortify the condition of our result.

1. Bruck R., On entire functions which share one value CM with their first derivative, Results Math., 30 (1996), 21-24.

2. Cao T.B., On the Bruck conjecture, Bull. Aust. Math. Soc., 93 (2016), 248-259.

3. Chen Z.X., Shon K.H., On conjecture of R. Bruck concerning the entire function sharing one value CM with its derivative, Taiwanese J. Math., 8 (2004), №2, 235-244.

4. Clunie J., On integral and meromorphic functions, J. London Math. Soc., 37 (1962), 17-22.

5. Gundersen G.G., Yang L.Z., Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl., 223 (1998), 88-95.

6. Hayman W.K., Meromorphic Functions, Clarendon Press, Oxford, 1964.

7. Lu F., Yi H.X., The Bruck conjecture and entire functions sharing polynomials with their k-th derivatives, J. Korean Math. Soc., 48 (2011), №3, 499-512.

8. Lu F., Li Q., Yang C., On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc., 51 (2014), №5, 1281-1289.

9. Yang L.Z., Zhang J.L., Non-existence of meromorphic solutions of Fermat type functional equation, Aequations Math., 76 (2008), №1-2, 140-150.

10. Yang C.C., On deficiencies of differential polynomials II, Math. Z., 125 (1972), 107-112.

11. Zhang J.L., Meromorphic functions sharing a small function with their derivatives, Kyungpook Math. J., 49 (2009), 143-154.

12. Zhang J.L., Yang L.Z., A power of a meromorphic function sharing a small function with its derivative, Annales Academiae Scientiarum Fennicae Mathematica, 34 (2009), 249-260.

13. Zhang J.L., Yang L.Z., A power of an entire function sharing one value with its derivative, Comput. Math. Appl, 60 (2010), 2153-2160.

	On the transcendental meromorphic solutions of a certain class of differential equations
Author	S. Majumder¹, A. Dam² smajumder05@yahoo.in¹, sujoy.katwa@gmail.com, sm05math@gmail.com² Department of Mathematics, Raiganj University, Raiganj, West Bengal, India
Abstract	In this paper we consider the differential equation \[F^{(k)}-a_{2}=e^{\gamma}\{\alpha(F-a_{1})+\beta\},\] where $a_{i}(z), \alpha(z)(\not\equiv 0,\infty)$, $\beta(z)(\not\equiv \infty)$, $i=1,2$ are small functions of $F$, $\gamma$ is an entire function and $k\in\mathbb{N}$. Let $f$ be a transcendental meromorphic function such that $ N(r,\infty;f)=S(r,f)$ and $n\in\mathbb{N}$ such that $n\geq k+1$. If $F=f^{n}$ is a solution of the above differential equation, then \[F^{(k)}\equiv \frac{a_{2}\alpha}{a_{1}\alpha-\beta}F.\] Also we exhibit an example to fortify the condition of our result.
Keywords	meromorphic functions; derivative; small function
DOI	doi:10.15330/ms.51.2.130-142
Reference	1. Bruck R., On entire functions which share one value CM with their first derivative, Results Math., 30 (1996), 21-24. 2. Cao T.B., On the Bruck conjecture, Bull. Aust. Math. Soc., 93 (2016), 248-259. 3. Chen Z.X., Shon K.H., On conjecture of R. Bruck concerning the entire function sharing one value CM with its derivative, Taiwanese J. Math., 8 (2004), №2, 235-244. 4. Clunie J., On integral and meromorphic functions, J. London Math. Soc., 37 (1962), 17-22. 5. Gundersen G.G., Yang L.Z., Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl., 223 (1998), 88-95. 6. Hayman W.K., Meromorphic Functions, Clarendon Press, Oxford, 1964. 7. Lu F., Yi H.X., The Bruck conjecture and entire functions sharing polynomials with their k-th derivatives, J. Korean Math. Soc., 48 (2011), №3, 499-512. 8. Lu F., Li Q., Yang C., On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc., 51 (2014), №5, 1281-1289. 9. Yang L.Z., Zhang J.L., Non-existence of meromorphic solutions of Fermat type functional equation, Aequations Math., 76 (2008), №1-2, 140-150. 10. Yang C.C., On deficiencies of differential polynomials II, Math. Z., 125 (1972), 107-112. 11. Zhang J.L., Meromorphic functions sharing a small function with their derivatives, Kyungpook Math. J., 49 (2009), 143-154. 12. Zhang J.L., Yang L.Z., A power of a meromorphic function sharing a small function with its derivative, Annales Academiae Scientiarum Fennicae Mathematica, 34 (2009), 249-260. 13. Zhang J.L., Yang L.Z., A power of an entire function sharing one value with its derivative, Comput. Math. Appl, 60 (2010), 2153-2160.
Pages	130-142
Volume	51
Issue	2
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On the transcendental meromorphic solutions of a certain class of differential equations