Previous |  Up |  Next


entire functions; order of growth; shared values; uniqueness theorems
We prove a theorem on the growth of nonconstant solutions of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its linear differential polynomial share a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.
[1] Brück, R.: On entire functions which share one value CM with their first derivative. Results in Math. 30 (1996), 21-24. DOI 10.1007/BF03322176 | MR 1402421
[2] Chen, Z. X., Yang, C. C.: Some further results on the zeros and growths of entire solutions of second order linear differential equation. Kodai Math J. 22 (1999), 273-285. DOI 10.2996/kmj/1138044047 | MR 1700597
[3] Chen, Z. X.: The growth of solutions of $f''+{\rm e}^{-z}f'+Q(z)f=0$. Science in China (A) 31 (2001), 775-784.
[4] Chang, J. M., Fang, M. L.: Entire functions that share a small function with their derivatives. Complex Variables Theory Appl. 49 (2004), 871-895. DOI 10.1080/02781070410001732205 | MR 2101212 | Zbl 1069.30049
[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. DOI 10.1006/jmaa.1998.5959 | MR 1627356 | Zbl 0911.30022
[6] Hayman, W. K.: Meromorphic Functions. The Clarendon Press Oxford (1964). MR 0164038 | Zbl 0115.06203
[7] He, Y. Z., Xiao, X. Z.: Algebroid Functions and Ordinary Differential Equations. Science Press Beijing (1988), Chinese.
[8] Jank, G., Volkmann, L.: Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen. Birkhäuser Basel-Boston (1985). MR 0820202 | Zbl 0682.30001
[9] Laine, I.: Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter Berlin (1993). MR 1207139
[10] Ngoan, V., Ostrovskij, I. V.: The logarithmic derivative of a meromorphic function. (Russian). Dokl. Akad. Nauk Arm. SSR 41 (1965), 272-277. MR 0192057
[11] Rubel, L., Yang, C. C.: Values shared by an entire function and its derivative. in``Complex Analysis, Kentucky 1976'' (Proc. Conf.), Lecture Notes in Mathematics, Vol. 599, pp. 101-103, Springer-Verlag, Berlin, 1977. MR 0460640 | Zbl 0362.30026
[12] Valiron, G.: Lectures on the General Theory of Integral Functions. Edouarard Privat Toulouse (1923) JFM 50.0254.01.
[13] Wang, J. P.: Entire functions that share a polynomial with one of their derivatives. Kodai Math J. 27 (2004), 144-151. DOI 10.2996/kmj/1093351321 | MR 2069765 | Zbl 1067.30067
[14] Yang, L.: Value Distribution Theory. Springer-Verlag Berlin Heidelberg (1993). MR 1301781 | Zbl 0790.30018
[15] Yang, C. C., Yi, H. X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers Dordrecht/Boston/London (2003). MR 2105668 | Zbl 1070.30011
[16] Yang, L. Z.: Solution of a differential equation and its applications. Kodai Math J. 22 (1999), 458-464. DOI 10.2996/kmj/1138044097 | MR 1727305 | Zbl 1004.30021
[17] Chuang, C. T.: Sur la comparaison de la croissance d'une fonction méromorphe et celle de sa dérivée. Bull. Sci. Math. 75 (1951), 171-190. MR 0045822
Partner of
EuDML logo