Abstract
Let f(z) be a transcendental meromorphic function, whose zeros have multiplicity at least 3. Set α(z): = β(z)exp (γ(z), where β(z) is a nonconstant elliptic function and γ(z) is an entire function. If σ(f(z)) > σ(α(z)), then f′(z) = α(z) has infinitely many solutions in the complex plane.
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References
W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. Math., 70 (1959), 9–42.
X. J. Liu, S. Nevo and X. C. Pang, On the kth derivative of meromorphic functions with zeros of multiplicity at least k + 1, J. Math. Anal. Appl., 348 (2008), 516–529.
S. Nevo, On theorems of Yang and Schwick, Complex Var. Elliptic Equ., 46 (2001), 315–321.
S. Nevo, Applications of Zalcman’s lemma to Qm-normal families, Analysis (Munich), 21 (2001), 289–325.
X. C. Pang, S. Nevo and L. Zalcman, Derivatives of meromorphic functions with multiple zeros and rational functions, Comput. Methods Funct. Theory, 8 (2008), 483–491.
X. C. Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc., 32 (2000), 325–331.
M. Tsuji, On Borel’s directions of meromorphic functions of finite order. II, Kodai Math. Sem. Rep., 2 (1950), 96–100.
Y. F. Wang and M. L. Fang, Picard values and normal families of meromorphic functions with multiple zeros, Acta Math. Sin., 14 (1998), 17–26.
Y. Xu, Normal families and exceptional functions, J. Math. Anal. Appl., 329 (2007), 1343–1354.
K. Yamanoi, The second main theorem for small functions and related problems, Acta Math., 192 (2004), 225–294.
P. Yang, Quasinormal criterion and Picard type theorem, Acta Math. Sci. (Ser. A), 35A (2015), 1089–1105.
P. Yang, P. Y. Niu and X. C. Pang, A Picard type theorem concerning meromorphic functions of hyper-order, Sci. Sin. Math., 47 (2017), 357–370 (in Chinese).
L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc., 35 (1998), 215–230.
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The author would like to thank the referees for valuable remarks and suggestions.
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 11671191, 11871216.
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Yang, P., Wang, S. On the Distribution of Meromorphic Functions of Positive Hyper-Order. Anal Math 47, 243–260 (2021). https://doi.org/10.1007/s10476-021-0070-1
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DOI: https://doi.org/10.1007/s10476-021-0070-1