Abstract
Define the differential operators ϕn for n ∈ ℕ inductively by ϕ1 [f](z)= f (z) and \({\phi _{n + 1}}[f](z) = f(z){\phi _n}[f](z) + {d \over {dz}}{\phi _n}[f](z)\). For a positive integer k ≥ 2 and a positive number δ, let \({\cal F}\) be the family of functions f meromorphic on domain D ⊂ ℂ such that ϕk[f](z) ≠ 0 and ∣Res(f, a) − j∣ ≥ δ for all j ∈{0, 1,…,k − 1} and all simple poles a of f in D. Then \({\cal F}\) is quasi-normal on D of order 1.
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References
Bergweiler, W.: Normality and exceptional values of derivatives. Proc. Amer. Math. Soc., 129, 121–129 (2001)
Bergweiler, W.: Bloch’s principle. Comput. Methods Funct. Theory, 6, 77–108 (2006)
Bergweiler, W., Langley, J. K.: Nonvanishing derivatives and normal families. J. Anal. Math., 91, 353–367 (2003)
Chang, J. M.: On meromorphic functions whose first derivatives have finitely many zeros. Bull. London Math. Soc., 44(4), 703–715 (2012)
Clunie, J.: On integral and meromorphic functions. J. London Math. Soc., 37, 17–27 (1962)
Chuang, C. T.: Normal Families of Meromorphic Functions, World Scientific, River Edge, NJ, 1993
Frank, G.: Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen. Math. Z., 149, 29–36 (1976)
Hayman, W. K.: Picard values of meromorphic functions and their derivatives. Ann. Math., 70, 9–42 (1959)
Hayman, W. K.: Meromorphic Functions, Clarendon Press, Oxford, 1964
Langley, J. K.: Proof of a conjecture of Hayman concerning f and f″. J. London Math. Soc., 48, 500–514 (1993)
Pang, X. C., Zalcman, L.: Normal families and shared values. Bull. London Math. Soc., 32, 325–331 (2000)
Pang, X. C., Nevo, S., Zalcman, L.: Quasinormal families of meromorphic functions. Rev. Mat. Ibero., 21(1), 249–262 (2005)
Schiff, J.: Normal Families, Springer-Verlag, Berlin, 1993
Schwick, W.: Normality criteria for families of meromorphic functions. J. Anal. Math., 52, 241–289 (1989)
Yang, L.: Value Distribution Theory, Springer-Verlag, Berlin, 1993
Zalcman, L.: Normal families: new perspectives. Bull. Amer. Math. Soc., (N. S.), 35, 215–230 (1998)
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Research Supported by NSFC (Grant No. 11471163)
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Chang, J.M. Quasi-normal Family of Meromorphic Functions Whose Certain Type of Differential Polynomials Have No Zeros. Acta. Math. Sin.-English Ser. 37, 1267–1277 (2021). https://doi.org/10.1007/s10114-021-0328-3
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DOI: https://doi.org/10.1007/s10114-021-0328-3