Value Distribution of the *k*th Derivatives of Meromorphic Functions

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In the paper, we take up a
new method to prove a result of value distribution of meromorphic functions:
let *f* be a meromorphic function in , and let , where *P* is a polynomial. Suppose that all zeros
of *f* have multiplicity at least , except possibly finite many, and as . Then has infinitely many zeros.

References

[1] W. K. Hayman, “Picard Values of Meromorphic Functions and Their Derivatives,” Annals of Mathematics, Vol. 70, No. 1, 1959, pp. 9-42. http://dx.doi.org/10.2307/1969890

[2] X. J. Liu, S. Nevo and X. C. Pang, “On the kth Derivative of Meromorphic Functions with Zeros of Multiplicity at Least k+1,” Journal of Mathematical Analysis and Applications, Vol. 348, No. 1, 2008, pp. 516-529.

http://dx.doi.org/10.1016/j.jmaa.2008.07.019

[3] S. Nevo, X. C. Pang and L. Zalcman, “Quasinormality and meromorphic functions with multiple zeros,” Journal d’Analyse Math??matique, Vol. 101, No. 1, 2007, pp. 1-23.

[4] X. C. Pang, S. Nevo and L. Zalcman, “Derivatives of Meromorphic Functions with Multiple Zeros and Rational Functions,” Computational Methods and Function Theory, Vol. 8, No. 2, 2008, pp. 483-491.

http://dx.doi.org/10.1007/BF03321700

[5] X. C. Pang and L. Zalcman, “Normal Families and Shared Values,” Bulletin London Mathematical Society, Vol. 32, No. 3, 2000, pp. 325-331. http://dx.doi.org/10.1112/S002460939900644X