In this paper, the authors prove a big Picard type theorem concerning derivative: Let f(z) be meromorphic in D = {z: 0 < ∣z − z0∣ < δ} for each δ > 0, if z0 is an essential singularity of f(z), then either f(z) assumes every finite value infinitely often or f′(z) assumes every finite value except possibly zero infinitely often. As an application of this result, they extend Nevo, Pang and Zalcman’s quasinormal criterion: Let {fn(z)} be a sequence of meromorphic functions on the plane domain D, all of whose zeros are multiple such that f′n(z) − 1 has zeros with multiplicity at least n for all n on D, then {fn(z)} is quasinormal of order 1 on D. Then they obtain a corresponding result in value distribution theory: Let f(z) be a meromorphic function on ℂ, all but finitely many of whose zeros are multiple such that<Equation ID="Equa"> <EquationSource Format="TEX">$\mathop {\overline {\lim}}\limits_{r \to + \infty} {{T({r,f})} \over {{{({\log r})}^2}}} = + \infty,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <munder> <mrow> <mover> <mo form="prefix">lim</mo> <mo accent="false">¯</mo> </mover> </mrow> <mrow> <mi>r</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mfrac> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mrow> <mi>r</mi> <mo>,</mo> <mi>f</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mrow> <mi>log</mi> <mi>r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </math></EquationSource> </Equation> then there exist a positive integer M and R0 > 0 such that for each r > R0, there exists z0 ∈ ℂ satisfying ∣z0∣ > r such that z0 is a zero of f′(z) − 1 with multiplicity at most M.

A Big Picard Type Theorem Concerning Derivative and Its Application

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发表日期：

2025年11月7日

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Shuxian Li,
Xiaojun Liu

摘要

In this paper, the authors prove a big Picard type theorem concerning derivative: Let f(z) be meromorphic in D = {z: 0 < ∣z − z₀∣ < δ} for each δ > 0, if z₀ is an essential singularity of f(z), then either f(z) assumes every finite value infinitely often or f′(z) assumes every finite value except possibly zero infinitely often. As an application of this result, they extend Nevo, Pang and Zalcman’s quasinormal criterion: Let {f_n(z)} be a sequence of meromorphic functions on the plane domain D, all of whose zeros are multiple such that f′_n(z) − 1 has zeros with multiplicity at least n for all n on D, then {f_n(z)} is quasinormal of order 1 on D. Then they obtain a corresponding result in value distribution theory: Let f(z) be a meromorphic function on ℂ, all but finitely many of whose zeros are multiple such that $\mathop {\overline {\lim}}\limits_{r \to + \infty} {{T({r,f})} \over {{{({\log r})}^2}}} = + \infty,$ $\underset{r \to + \infty}{lim^{¯}} \frac{T (r, f)}{{(\log r)}^{2}} = + \infty,$ then there exist a positive integer M and R₀ > 0 such that for each r > R₀, there exists z₀ ∈ ℂ satisfying ∣z₀∣ > r such that z₀ is a zero of f′(z) − 1 with multiplicity at most M.