Abstract
In this paper, the normality of a family of meromorphic functions is deduced from the normality of a given family. Precisely, we have proved: Let \({\mathcal {F}}\) and \({\mathcal {G}}\) be two families of meromorphic functions on a domain D, and \(a,\ b,\ c\) be three finite complex numbers such that \(a\ne 0\) and \(b\ne c\). Suppose that \({\mathcal {G}}\) is normal in D such that no sequence in \({\mathcal {G}}\) converges locally uniformly to infinity in D. If \(n\ge 3\) and for each function \(f\in {\mathcal {F}}\) there exists \(g\in {\mathcal {G}}\) such that \(f^{'}-af^{n}\) and \(g^{'}-ag^{n}\) partially share the values b and c, then \({\mathcal {F}}\) is normal in D. Further, examples are given to establish the sharpness of the result.
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1 Introduction and Main Results
Let D be a domain in the complex plane \({\mathbb {C}}\). A family \({\mathcal {F}}\) of meromorphic functions on D is said to be normal if from every sequence \(\{f_n\}\) in \({\mathcal {F}},\) we can extract a subsequence \(\{f_{n_k}\}\) which converges locally uniformly to f in D with respect to the spherical metric, where f is either a meromorphic function or identically equal to infinity in D. A family \({\mathcal {F}}\) is said to be normal at \(z_0\in D\) if it is normal in some neighborhood of \(z_0\); thus, \({\mathcal {F}}\) is normal in D if and only if it is normal at each point \(z\in D\). (see [14]).
Let f and g be two meromorphic functions in D and let \(a\in {\mathbb {C}}\). We shall denote by E(f, a) the set of zeros of \(f-a\) (ignoring multiplicities). We say that f and g share the value a if \(E(f,a)=E(g,a)\). Further, if \(E(f,a)\subset E(g,a)\), we say that f and g share the value a partially (see [18]).
According to Bloch’s principle [14], any condition which reduces a meromorphic function in \({\mathbb {C}}\) to a constant is likely to force a family of meromorphic functions in a domain D to be normal. Although this principle as well as its converse does not hold in general (see, for example [2, 13]), still it serves as a guiding principle for obtaining normality criteria corresponding to Picard-type theorems and vice versa (see [1]).
In 1959, Hayman [5] proved that if f is a meromorphic function in the complex plane, \(a \in {\mathbb {C}}{\setminus } \{0\}\) and the differential polynomial \(f^{'}-af^n, \ n\ge 5\), does not assume a finite complex value in \({\mathbb {C}}\), then f is constant. This result is not true for \(n=3,4\) as shown by Mues [10]. In view of Bloch’s principle, Hayman [6] in 1967 conjectured that there exists a normality criterion corresponding to this Picard-type theorem. Over the next few decades, the following normality criterion was established thereby proving the Hayman’s conjecture.
Theorem 1.1
Let \({\mathcal {F}}\) be a family of meromorphic (holomorphic) functions in a domain D, \(n\in {\mathbb {N}}\) and a, b be two finite complex numbers such that \(n\ge 3 \ (n\ge 2)\) and \(a\ne 0\). If for each \(f\in {\mathcal {F}}\), \(f^{'}-af^n\ne b\), then \({\mathcal {F}}\) is normal in D.
The proof of Theorem 1.1 for meromorphic functions is due to S. Li [8], X. Li [9] and Langley [7] for \(n\ge 5\), Pang [11] for \(n=4\), Chen and Fang [3] and Zalcman [17] for \(n=3\) independently and the proof of Theorem 1.1 for holomorphic functions is due to Drasin [4] for \(n\ge 3\) and Ye [16] for \(n=2\).
In 2008, Zhang [19] considered the idea of shared values and proved the following.
Theorem 1.2
Let \({\mathcal {F}}\) be a family of meromorphic (holomorphic) functions in D, \(n\in {\mathbb {N}}\) and a, b be two finite complex numbers such that \(n\ge 4 \ (n\ge 2)\) and \(a\ne 0\). If for each pair of functions f and g in \({\mathcal {F}}\), \(f^{'}-af^n\) and \(g^{'}-ag^n\) share the value b, then \({\mathcal {F}}\) is normal in D.
In this paper, we consider the related problems concerning two families of meromorphic functions and prove the following theorem:
Theorem 1.3
Let \({\mathcal {F}}\) and \({\mathcal {G}}\) be two families of holomorphic functions on a domain D, and \(a,\ b,\ c\) be three complex numbers such that \(a\ne 0\) and \(b\ne c\). Suppose that \({\mathcal {G}}\) is normal in D such that no sequence in \({\mathcal {G}}\) converges locally uniformly to infinity in D. If \(n\ge 2\) and for each function \(f\in {\mathcal {F}}\), there exists \(g\in {\mathcal {G}}\) such that \(f^{'}-af^{n}\) and \(g^{'}-ag^{n}\) partially share the values b and c, then \({\mathcal {F}}\) is normal in D.
In the following example, we show that the condition ‘partial sharing of two values b and c’ in Theorem 1.3 cannot be reduced to one.
Example 1.4
Consider the two families \({\mathcal {F}}:=\left\{ f_j(z)=e^{jz}:j\in {\mathbb {N}}\right\} \) and \({\mathcal {G}}:=\left\{ 1\right\} \) of holomorphic functions on \({\mathbb {D}}\). Note that \(g_j^{'}-g_j^2\equiv -1\). Therefore, \(f_j^{'}-f_j^2= -1\Rightarrow g_j^{'}-g_j^2=-1\). But \({\mathcal {F}}\) fails to be normal at \(z=0\).
We demonstrate in the subsequent example that Theorem 1.3 fails to be true when \(n=1\). Therefore, the condition \(n=2\) is the best possible for Theorem 1.3.
Example 1.5
Consider the two families \({\mathcal {F}}:=\left\{ f_j(z)=jz:j\in {\mathbb {N}}\right\} \) and \({\mathcal {G}}:=\left\{ -1\right\} \) of holomorphic functions on \({\mathbb {D}}\). Then, clearly, \(f_j^{'}(z)-f_j(z)=j(1-z)\ne 0\), and for each \(f_j\in {\mathcal {F}}\), there exists \(g_j\in {\mathcal {G}}\) such that \(f_j^{'}(z)-f_j(z)=1\Rightarrow g_j^{'}(z)-g_j(z)=1\). But \({\mathcal {F}}\) fails to be normal at \(z=0\).
The following example illustrates that Theorem 1.3 is not valid for the family of meromorphic functions when \(n=2\).
Example 1.6
Consider the two families
and
of meromorphic functions on \({\mathbb {D}}\). Take \(a=-1\). Then, clearly, \(f_j^{'}(z)-af_j^2(z)=\frac{j}{(1+jz^2)^2}\ne 0\) and for each \(f_j\in {\mathcal {F}}\), there exists \(g_j\in {\mathcal {G}}\) such that \(f_j^{'}(z)-af_j^2(z)=1\Rightarrow g_j^{'}(z)-ag_j^2(z)=1\). But \({\mathcal {F}}\) is not normal at \(z=0\) since \(f_j(0)=0\) and for \(z\ne 0,\) \(f_j(z)\rightarrow 1/z\) as \(n\rightarrow \infty \).
However, Theorem 1.3 can be extended to families of meromorphic functions provided that \(n\ge 3\).
Theorem 1.7
Let \({\mathcal {F}}\) and \({\mathcal {G}}\) be two families of meromorphic functions on a domain D, and \(a,\ b,\ c\) be three finite complex numbers such that \(a\ne 0\) and \(b\ne c\). Suppose that \({\mathcal {G}}\) is normal in D such that no sequence in \({\mathcal {G}}\) converges locally uniformly to infinity in D. If \(n\ge 3\) and for each function \(f\in {\mathcal {F}}\), there exists \(g\in {\mathcal {G}}\) such that \(f^{'}-af^{n}\) and \(g^{'}-ag^{n}\) partially share the values b and c, then \({\mathcal {F}}\) is normal in D.
In the following example, we show that the condition ‘partial sharing of two values b and c’ in Theorem 1.7 cannot be reduced to one.
Example 1.8
Consider the two families
and
of meromorphic functions on \({\mathbb {D}}\). Then for each \(f_j\in {\mathcal {F}},\) there exists \(g_j\in {\mathcal {G}}\) such that \(f_j^{'}-f_j^3=0\Rightarrow g_j^{'}-g_j^3=0\). Also, \(g_j(z)\rightarrow g(z)=\frac{1}{z-1}\not \equiv \infty \). But \({\mathcal {F}}\) fails to be normal at \(z=0\).
For \(n=2\), we have the following weak version of the Theorem 1.7.
Theorem 1.9
Let \({\mathcal {F}}\) and \({\mathcal {G}}\) be two families of meromorphic functions on a domain D such that each \(f\in {\mathcal {F}}\) has neither simple zeros nor simple poles. Let \(a,\ b\) and c be three finite complex numbers such that \(a\ne 0\) and \(b\ne c\). Suppose that \({\mathcal {G}}\) is normal in D such that no sequence in \({\mathcal {G}}\) converges locally uniformly to infinity in D. If for each function \(f\in {\mathcal {F}}\), there exists \(g\in {\mathcal {G}}\) such that \(f^{'}-af^{2}\) and \(g^{'}-ag^{2}\) partially share the values b and c, then \({\mathcal {F}}\) is normal in D.
Note that Example 1.6 also shows that the condition ‘each \(f\in {\mathcal {F}}\) has neither simple zeros nor simple poles’ in Theorem 1.9 can not be omitted.
2 Lemmas and Proof of the Results
To prove our results, we need the following lemmas.
Lemma 2.1
[12] Let \(\mathcal {F}\) be a family of meromorphic functions on the unit disk \({\mathbb {D}}\) such that all the zeros of \(f\in {\mathcal {F}}\) are of multiplicity at least p and all the poles of \(f\in {\mathcal {F}}\) are of multiplicity at least q. Suppose that \({\mathcal {F}}\) is not normal at \(z_0\in D\). Then, for every \(\alpha \in (-p, q)\), there exist
-
(a)
points \(z_n\) in \({\mathbb {D}}:\) \(z_n\rightarrow z_0;\)
-
(b)
functions \(f_n\in \mathcal {F};\)
-
(c)
positive real numbers \(\rho _n:\rho _n\rightarrow 0\)
such that the re-scaled sequence \(\left\{ g_n(\zeta )=\rho _{n}^{\alpha }f_n(z_n+\rho _n\zeta )\right\} \) converges spherically locally uniformly on \(\mathbb {C}\) to a non-constant meromorphic function g on \({\mathbb {C}}\) of finite order.
Lemma 2.2
[3] Let f be a meromorphic function in \({\mathbb {C}}\), and let n be a positive integer. If \(f^nf^{'}\) does not assume a non-zero finite complex number in \({\mathbb {C}}\), then f is constant.
Lemma 2.3
[15] Let f be a meromorphic function in \({\mathbb {C}}\) and b be a non-zero complex number. If f has neither simple zero nor simple pole and \(f^{'}(z)\ne b\), then f is constant.
Proof of the Theorem 1.3
We may consider D to be an open unit disk \({\mathbb {D}}.\) Suppose that the family \({\mathcal {F}}\) is not normal at \(z_0\in {\mathbb {D}}\). Then by Lemma 2.1, there exist points \(z_j\in {\mathbb {D}}\) with \(z_j\rightarrow z_0\), a sequence of positive numbers \(\rho _j\rightarrow 0\) and a sequence of functions \(f_j\in {\mathcal {F}}\) such that
is locally uniformly on \({\mathbb {C}}\), where F is a non-constant entire function of finite order.
From (2.1), we have
and
locally uniformly on \({\mathbb {C}}\).
For each \(f_j\in {\mathcal {F}}\), there exists \(g_j\in {\mathcal {G}}\) such that \(f_j^{'}-af_j^n\) and \(g_j^{'}-ag_j^n\) share the values b and c partially in \({\mathbb {D}}\). Since \({\mathcal {G}}\) is normal, there exists a subsequence in \(\{g_j\}\), again denoted by \(\{g_j\}\), that converges uniformly to a holomorphic function \(g(z)\not \equiv \infty \) in some neighborhood of \(z_0\).
Suppose \((F^{'}-aF^{n})\not \equiv 0\) otherwise \(\frac{-1}{n-1}\frac{1}{F^{n-1}}\equiv a\zeta + d\), for some \(d\in {\mathbb {C}}\), which contradicts to the fact that F is an entire function and \(n\ge 2\). Further, suppose that \((F^{'}-aF^{n})(\zeta )\ne 0\), \(\zeta \in {\mathbb {C}}\). Then \(\frac{F^{'}}{F^{n}}\ne a\). By setting \(F=1/\phi \), we have \(\phi ^{n-2}\phi ^{'}\ne -a\). When \(n\ge 3\), \(\phi \) is constant by Lemma 2.2 and when \(n=2\), \(\phi \) is again constant by Hayman’s alternative since \(\phi \ne 0\) and \(\phi ^{'}\ne -a\). In both cases, \(\phi \) is constant. This implies that F is constant, a contradiction. Thus, \((F^{'}-aF^{n})\) has at least one zero.
Now we have two cases:
Case-I. \((g^{'}-ag^{n})(z_0)\ne b\).
Suppose that \((F^{'}-aF^{n})(\zeta _0)= 0\), for some \(\zeta _0\in {\mathbb {C}}\). From (2.2), by Hurwitz’s theorem, there exists a sequence \(\{\zeta _j\}\) with \(\zeta _j\rightarrow \zeta _0\) such that for sufficiently large j
and thus
By hypothesis, we have \((g^{'}_j-ag^{n}_j)(z_j+\rho _j\zeta _j)=b\) and so \((g^{'}-ag^{n})(z_0)=b\), a contradiction.
Case-II. \((g^{'}-ag^{n})(z_0) = b.\)
By using (2.3) instead of (2.2) in Case-I, we obtain \((g^{'}-ag^{n})(z_0)=c\ (\ne b)\) which is not true. This completes the proof. \(\square \)
Proof of the Theorem 1.7
We may consider D to be an open unit disk \({\mathbb {D}}.\) Suppose that the family \({\mathcal {F}}\) is not normal at \(z_0\in {\mathbb {D}}\). Then there exists a sequence \(\{f_n\}\subset {\mathcal {F}}\) which has no locally convergent subsequence at \(z_0\). Thus, by Lemma 2.1, there exist points \(z_j\in {\mathbb {D}}\) with \(z_j\rightarrow z_0\), a sequence of positive numbers \(\rho _j\rightarrow 0\), and a sequence of functions in \(\{f_j\}\) again denoted by \(\{f_j\}\) such that
locally uniformly on \({\mathbb {C}}\) with respect to spherical metric, where F is a non-constant meromorphic function on \({\mathbb {C}}\) of finite order.
From (2.4), we have
and
spherically locally uniformly on \({\mathbb {C}}\) except possibly at the poles of F.
For each \(f_j\in {\mathcal {F}}\), there exists \(g_j\in {\mathcal {G}}\) such that \(f_j^{'}-af_j^n\) and \(g_j^{'}-ag_j^n\) partially share the values b and c in \({\mathbb {D}}\). Since \({\mathcal {G}}\) is normal, there exists a subsequence in \(\{g_j\}\), again denoted by \(\{g_j\}\), that converges uniformly to a meromorphic function \(g(z)\not \equiv \infty \) in some neighborhood of \(z_0\).
Claim. \((F^{'}-aF^{n})(\zeta _0)=0\), for some \(\zeta _0\in {\mathbb {C}}\).
Suppose that \((F^{'}-aF^{n})(\zeta )\ne 0\). Then \(\frac{F^{'}}{F^{n}}\ne a\). By setting \(F=1/\phi \), \(\phi ^{n-2}\phi ^{'}\ne -a\). By Lemma 2.2, \(\phi \) and so F is constant, a contradiction. This proves the claim.
Now we have three cases:
Case-I. \((g^{'}-ag^{n})(z_0)\ne b,\infty \).
By Claim, \((F^{'}-aF^{n})(\zeta _0)= 0\), for some \(\zeta _0\in {\mathbb {C}}\). Since \((F^{'}-aF^{n})\not \equiv 0\), otherwise \(\frac{-1}{n-1}\frac{1}{F^{n-1}}\equiv a\zeta + d\), for some \(d\in {\mathbb {C}}\), which contradicts to the fact that F is a non-constant meromorphic function and \(n\ge 3\), by (2.5), there exists a sequence \(\{\zeta _j\}\) with \(\zeta _j\rightarrow \zeta _0\) such that for sufficiently large j, \((f^{'}_j-af^{n}_j)(z_j+\rho _j\zeta _j)=b.\) By assumption, we have \((g^{'}_j-ag^{n}_j)(z_j+\rho _j\zeta _j)=b\) and so \((g^{'}-ag^{n})(z_0)=b\), a contradiction.
Case-II. \((g^{'}-ag^{n})(z_0) = b.\)
Using (2.6) instead of (2.5) in Case-I, we obtain \((g^{'}-ag^{n})(z_0)=c\ (\ne b)\), which is not true.
Case-III. \((g^{'}-ag^{n})(z_0) = \infty .\)
Then, clearly, \(g(z_0)=\infty \). Suppose that \(z_0\) is a pole of g with multiplicity \(k\ge 1\). Then, for sufficiently large j, \(g_j\) has exactly \(l\le k\) distinct poles \(z_j^1,\ldots , z_j^l\) in \(D(z_0,r)\) with multiplicities \(\alpha _1,\ldots ,\alpha _l\) respectively such that \(z_j^i\rightarrow z_0\ (i=1,\ldots , l)\) and \(\sum _{i=1}^{l}\alpha _i=k\). Renumbering if possible, we may assume that the number l and multiplicities \(\alpha _i, i=1,\ldots , l\) are independent of j. Now set
Then the functions \(H_n\) are holomorphic in \(D(z_0,r)\) and \(H_n\rightarrow H\) on \(D(z_0,r/2)\setminus \{z_0\}\), where \(H(z)=g(z)(z-z_0)^k\) is holomorphic on \(D(z_0,r)\). Note that \(H(z_0)\ne 0, \infty .\) Hence by maximum principle, \(H_n\rightarrow H\) on \(D(z_0,r/2)\).
We have
Then
where
Since \(H(z_0)\ne 0,\infty \), we have
and
Therefore, \(K_j(z)\) and so \(g_j^{'}(z)-ag_j^n(z)-b\) has no zeros in some neighborhood of \(z_0\). By assumption, we find that \(f_j^{'}(z)-af_j^n(z)-b\) has no zero in some neighborhood of \(z_0\). By Theorem 1.1, the sequence \(\{f_j\}\) is normal at \(z_0\), a contradiction. \(\square \)
Proof of the Theorem 1.9
Following the proof of Theorem 1.7, we only need to prove that \(F^{'}-aF^{2}\not \equiv 0\) and \(F^{'}-aF^{2}\) has at least one zero. Suppose that \(F^{'}-aF^{2}\equiv 0\). Then \((\frac{1}{F})^{'}\equiv a\) which implies that \(\frac{1}{F}\equiv a\zeta + d\), for some \(d\in {\mathbb {C}}\), which contradicts the fact that F has no simple pole. Next, suppose that \(F^{'}-aF^{2}\ne 0\). Then \(\frac{F^{'}}{F^{2}}\ne a\). We set \(F=1/\phi \), \(\phi ^{'}\ne -a\). By Lemma 2.3, \(\phi \) and so F is a constant, a contradiction. \(\square \)
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Kumar, M. Normality of Two Families of Meromorphic Functions Concerning Partially Shared Values. Arnold Math J. 10, 257–264 (2024). https://doi.org/10.1007/s40598-023-00236-3
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DOI: https://doi.org/10.1007/s40598-023-00236-3