The Zeros of Complex Delay-differential Polynomials Related to Hayman Conjecture

Ying Chun, GAO Kai LIU

Acta Mathematica Sinica, Chinese Series ›› 2023, Vol. 66 ›› Issue (1) : 67-74.

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Acta Mathematica Sinica, Chinese Series ›› 2023, Vol. 66 ›› Issue (1) : 67-74. DOI: 10.12386/A20210079

The Zeros of Complex Delay-differential Polynomials Related to Hayman Conjecture

  • Ying Chun, GAO Kai LIU
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Abstract

This paper is based on the Hayman Conjecture for the zeros problem of complex differential polynomials. Using an important estimate on the zeros of higher derivative of meromorphic functions given by Yamanoi, we obtain the improved results on the zeros of delay-differential polynomials. For instance, we have that if f is a transcendental meromorphic function with hyper-order less than one and q ≥ p+s+t+1, then [Q(f)P(f(z +c))](k) -a has infinitely many zeros, where a is a non-zero constant, P(z) is a polynomial of degree p with t different zeros and Q(z) is a polynomial of degree q with s different zeros. Our results improve the former results which obtained mainly by the second main theorem of Nevanlinna.

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Nevanlinna theory / Hayman Conjecture / complex delay-differential polynomials / zeros

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Ying Chun, GAO Kai LIU. The Zeros of Complex Delay-differential Polynomials Related to Hayman Conjecture. Acta Mathematica Sinica, Chinese Series, 2023, 66(1): 67-74 https://doi.org/10.12386/A20210079

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