- Research
- Open access
- Published:
Characteristic estimation of differential polynomials
Journal of Inequalities and Applications volume 2021, Article number: 181 (2021)
Abstract
In this paper, we give the characteristic estimation of a meromorphic function f with the differential polynomials \(f^{l}(f^{(k)})^{n}\) and obtain that
holds for \(M=\min \{\frac{1}{l-2},6\}\), integers \(l(\geq 2)\), \(n(\geq 1)\), \(k(\geq 1)\), and a non-zero constant a. This quantitative estimate is an interesting and complete extension of earlier results. The value distribution of a differential monomial of meromorphic functions is also investigated.
1 Introduction and main results
We assume that the reader is familiar with the fundamentals of Nevanlinna’s value distribution theory of meromorphic functions (see e.g. [4, 10, 16]). Let f be a transcendental meromorphic function in the complex plane \(\mathbb{C}\). We denote by \(S(r,f)\) any quantity satisfying \(S(r,f)=o(T(r,f))\), as \(r\rightarrow \infty \), possibly outside of an exceptional set of finite logarithmic measure. A meromorphic function α defined in \(\mathbb{C}\) is called a small function of f if \(T(r,\alpha )=S(r,f)\).
We also introduce some other symbols (see [15]). Let \(a\in \mathbb{C}\cup \{\infty \}\), k be a positive integer. Let \(N_{k)} (r,\frac{1}{f-a} )\) denote the counting function of those a-points of f (counting multiplicity) whose multiplicities are not greater than k, and let \(\overline{N}_{k)} (r,\frac{1}{f-a} )\) denote the corresponding reduced counting function. Similarly, let \(N_{(k} (r,\frac{1}{f-a} )\) denote the counting function of those a-points of f (counting multiplicity) whose multiplicities are not less than k, and let \(\overline{N}_{(k} (r,\frac{1}{f-a} )\) denote the corresponding reduced counting function. And let \(N_{k} (r,\frac{1}{f-a} )\) denote the counting function of those a-points of f with multiplicity k.
Hayman [5] proved the following well-known theorem.
Theorem 1.1
([5, Theorem 9])
Let f be a transcendental meromorphic function in the complex plane, and let l be a positive integer. If \(l\geq 3\), then \(f^{l}f'\) assumes every finite nonzero value infinitely often.
Hayman also conjectured that Theorem 1.1 remained valid for \(l\geq 1\). Mues [12] proved that \(f^{2}f'-1\) has infinitely many zeros. Later on, many researchers investigated the zeros of differential monomial \(f^{l}(f^{(k)})^{n}-a\) for positive integers \(l,n,k\) and a non-zero complex number a, and obtained some qualitative results, see e.g. [2, 3, 11, 13, 14], and some quantitative results, see e.g. [1, 6–9, 17].
Zhang [17] proved that the inequality \(T(r,f)<6N (r,\frac{1}{f^{2}f'-1} )+S(r,f)\) holds. Huang and Gu [6] extended the inequality and proved the following.
Theorem 1.2
([6, Theorem 1])
Let f be a transcendental meromorphic function in the complex plane, and let k be a positive integer. Then
Karmakar and Sahoo further [8] proved the following.
Theorem 1.3
([8, Theorem 1.1])
Let f be a transcendental meromorphic function and \(l(\geq 2)\), \(k(\geq 1)\) be any integers, then
Lahiri and Dewan [9] obtained the following estimate.
Theorem 1.4
([9, Theorem 3.2])
Let f be a transcendental meromorphic function, \(\alpha (\not \equiv 0,\infty )\) be a small function of f. If \(\psi =\alpha f^{l}(f^{(k)})^{n}\), where \(l(\geq 0)\), \(n(\geq 1)\), \(k(\geq 1)\) are integers, then for any small function \(a(\not \equiv 0,\infty )\) of ψ,
where \(N_{(k)} (r,\frac{1}{f} )\) denotes the counting function of zeros of f, a zero with multiplicity q is counted q times if \(q\leq k\) and is counted k times if \(q>k\).
Remark 1.1
Estimate (3) implies that, for \(l\geq 3\), \(n\geq 1\), \(k\geq 1\),
Jiang and Huang [7] proved the following.
Theorem 1.5
([7, Theorem 1])
Let f be a transcendental meromorphic function in the complex plane, \(l(\geq 2)\), \(n(\geq 2)\), \(k(\geq 2)\) be integers, and a be a non-zero constant. Then
where \(S^{*}(r,f)\) denotes the quantity satisfying \(S^{*}(r,f)=o(T(r,f))\) for all r outside a possible exceptional set E of logarithmic density 0.
We note that Theorem 1.3 does not hold for \(n\geq 2\), Theorem 1.4 is invalid for \(l=2\), and Theorem 1.5 remains invalid for \(l=2, n=1, k=1\). Thus, by using a method different from the previous proofs, we continue to consider the characteristic estimate of more general forms \(f^{l}(f^{(k)})^{n}-a\) for a non-zero constant a, integers \(l\geq 2\), \(n\geq 1\), and \(k\geq 1\), and obtain its quantitative result as follows.
Theorem 1.6
Let f be a transcendental meromorphic function with finite order in the complex plane, \(l(\geq 2)\), \(n(\geq 1)\), \(k(\geq 1)\) be integers, and a be a non-zero constant. Then
for \(M=\min \{\frac{1}{l-2},6\}\).
The quantity
is called the deficiency of f at the point a. It is obvious that \(0\leq \Theta (a,f)\leq 1\). Thus, we present the value distribution of a differential monomial \(f^{l}(f^{(k)})^{n}\).
Corollary 1.1
Let f be a transcendental meromorphic function with finite order in the complex plane, \(l(\geq 2)\), \(n(\geq 1)\), \(k(\geq 1)\) be integers, and a be a non-zero constant. Then
for \(M=\min \{\frac{1}{l-2},6\}\).
2 Some lemmas
We now prepare some lemmas.
Lemma 2.1
Let f be a transcendental meromorphic function with finite order. Then \(f^{l}(f^{(k)})^{n}\) is not identically constant, where \(l(\geq 2)\), \(n(\geq 1)\), \(k(\geq 1)\) are integers.
Proof
Contradicting to our assumption, we suppose that \(f^{l}(f^{(k)})^{n}\equiv C\). Clearly, \(C\neq 0\). Then \(\frac{1}{f^{n+l}}=\frac{1}{C} (\frac{f^{(k)}}{f} )^{n}\), and
a contradiction. □
Lemma 2.2
Let f be a transcendental meromorphic solution with finite order. Suppose that \(g(z)=f^{2}(f^{(k)})^{n}-a\), where \(n(\geq 1)\), \(k(\geq 1)\) are integers and a is a non-zero constant. Then
and
where \(N_{0} (r,\frac{1}{g'} )\) denotes the counting function of those zeros of \(g'\), but not zeros of f or g.
Proof
It follows from Lemma 2.1 that g is not identically constant. Thus
We conclude from the lemma of the logarithmic derivative that
and
Denote
where \(N_{000} (r,\frac{1}{g'} )\) denotes the counting function of those zeros of \(g'\) which are from the zeros of g, \(N_{00} (r,\frac{1}{g'} )\) denotes the counting function of those zeros of \(g'\) which are from the zeros of f, \(N_{0} (r,\frac{1}{g'} )\) denotes the counting function of those zeros of \(g'\) which are not zeros of f or g. So, we have
Let \(z_{0}\) be a zero of f with multiplicity q. If \(q\leq k\), then \(z_{0}\) is a zero of \(g'\) with multiplicity at least \(2q-1\). If \(q\geq k+1\), then \(z_{0}\) is a zero of \(g'\) with multiplicity \((n+2)q-(nk+1)\). Thus, by simple calculation, we have
and
Then we deduce from (9)−(12) that
and so inequality (7) is proved.
We further get from (7) that
that is,
and so inequality (8) is proved. □
Lemma 2.3
Let f be a transcendental meromorphic function with finite order. Suppose that
where \(n(\geq 1)\), \(k(\geq 1)\) are integers and a is a non-zero constant.
where \(a_{i}'s\) are defined by
when \(k=1\), and are defined by
when \(k\geq 2\). Then \(F(z)\not \equiv 0\).
Proof
We use a similar method of Huang-Gu [6, Lemma 3]. Suppose that \(F(z)\equiv 0\), we claim that
-
(i)
\(g(z)\neq 0\);
-
(ii)
\(h(z)\neq 0\);
-
(iii)
all zeros of \(f(z)\) are simple.
Suppose that \(z_{1}\) is a zero of \(g(z)\) with multiplicity \(l(\geq 1)\). Then \(f(z_{1})\neq 0,\infty \), and \(z_{1}\) is a zero of \(h(z)\) with multiplicity \(l-1\) since \(g'=fh\). Using the Laurent series of \(F(z)\) at the point \(z_{1}\), we can calculate that the coefficient \(A(l)\) of \((z-z_{1})^{-2}\) is
Using (14) for \(k=1\), we have
This shows that \(z_{1}\) is a pole of \(F(z)\), which contradicts \(F(z)\equiv 0\). Hence \(g(z)\neq 0\) when \(k=1\). Using (15) for \(k\geq 2\), we have
Clearly, \(A(l)\neq 0\) for all positive integers l. This shows that \(z_{1}\) is again a pole of \(F(z)\), which contradicts \(F(z)\equiv 0\). Hence \(g(z)\neq 0\) when \(k\geq 2\).
Suppose that \(z_{2}\) is a zero of \(h(z)\) with multiplicity \(l(\geq 1)\). By \((i)\) we have \(g(z_{2})\neq 0,\infty \). Using the Laurent series of \(F(z)\) at the point \(z_{2}\), we can get the coefficient \(B(l)\) of \((z-z_{1})^{-2}\) is
Using (14) for \(k=1\), we get
and so, the point \(z_{2}\) is again a pole of \(F(z)\), which contradicts \(F(z)\equiv 0\).
Using (15) for \(k\geq 2\), we get
and so, the point \(z_{2}\) is a pole of \(F(z)\), which contradicts \(F(z)\equiv 0\). Hence conclusion \((ii)\) \(h(z)\neq 0\) holds when \(k\geq 1\).
Noting that \(h(z)=\frac{g'(z)}{f(z)}=2f'(f^{(k)})^{n}+nf(f^{(k)})^{n-1}f^{(k+1)}\) and \((ii)\) \(h(z)\neq 0\), we can obtain \((iii)\).
Setting \(\phi (z):=\frac{h(z)}{g(z)}\), we conclude that \(\phi (z)\) is an entire function, all zeros of \(\phi (z)\) can occur only at multiple poles of \(f(z)\) and the following expressions hold:
First, we consider the case \(k\geq 2\). Substituting the above two equalities into (13) yields
Applying (15), we have \(a_{2}+a_{3}=(nk+n+4)^{2}(nk+n+6)\neq 0\). And so \(\phi \not \equiv 0\), otherwise \(\frac{g'}{g}=f\phi \equiv 0\), that is, \(g\equiv C\), which contradicts Lemma 2.1. Thus, it follows from (16) that
where \(\alpha _{1i}(z)\ (i=1,2,3)\) are differential polynomials of \(\frac{\phi '}{\phi }\).
Differentiating both sides of (17) gives
Applying the above equality to (17), we have
where \(\alpha _{2i}(z)\ (i=1,2,3,4)\) are differential polynomials of \(\frac{\phi '}{\phi }\). Continuing the above process, we get
where \(\alpha _{ki}(z)\ (i=1,2,\ldots,k+2)\) are differential polynomials of \(\frac{\phi '}{\phi }\).
Suppose that \(z_{3}\) is a simple zero of \(f(z)\). Together with (17), (18) and noting that \(\phi (z_{3})\neq 0,\infty \), we have
Substituting the above two equalities into the expressions of \(g(z)\) and \(h(z)\) yields
Combining the above two equalities and the expression of \(\phi (z):=\frac{h(z)}{g(z)}\), we get
Set \(U(z):=a\phi ^{n+2}(z)+2\alpha _{11}(z)\alpha _{k1}^{n}(z)\). We consider the following two cases.
Case 1. \(U(z)\not \equiv 0\). It follows from (19) and \((iii)\) that
Using (8) and noting that \(N (r,\frac{1}{g} )=0\), we have
It follows from (20)−(22) that
Applying (22) and (23), we get
a contradiction.
Case 2. \(U(z)\equiv 0\). By the expression of \(U(z)\), and noting that \(\alpha _{11}(z),\alpha _{k1}(z)\) are differential polynomials of \(\frac{\phi '}{\phi }\), we conclude that
and
Using (24), we conclude that \(\phi (z)\) is a polynomial or a constant. If \(\phi (z)\) is a polynomial, the left-hand side of (25) is a polynomial, and the right-hand side of (25) is a constant or rational function, a contradiction. So, \(\phi (z)\) is a constant. If \(\phi (z)\equiv 0\), then \(\frac{g'}{g}=f\phi \equiv 0\), that is, g is a constant, a contradiction. If \(\phi (z)\equiv C(C\neq 0)\), then we substitute this equality into (16) and get
Using (25) for \(k\geq 2\), we have \(a_{1}+a_{4}+a_{5}=-\frac{(nk+n+4)^{2}(nk+n+6)}{nk+n+2}\neq 0\) and \(a_{2}+a_{3}=(nk+n+4)^{2}(nk+n+6)\neq 0\). Thus \((\frac{1}{f} )'=-\frac{f'}{f^{2}}\equiv - \frac{C}{nk+n+2}\neq 0\), and f is a rational function, a contradiction.
We now consider the case \(k=1\). Similar to the proof of the case \(k\geq 2\), we obtain a contradiction. □
Lemma 2.4
Let \(f(z)\), \(g(z)\), \(h(z)\), and \(F(z)\) be stated as in Lemma 2.3. Then all simple poles of \(f(z)\) are zeros of \(F(z)\).
Proof
Suppose that \(z_{0}\) is a simple pole of \(f(z)\), then
where \(A\neq 0\), \(b,c\) are constants. We consider the following two cases.
Case 1. \(k=1\). We have
By using the above two equalities, we have
By substituting the above equalities into (13) and performing some easy calculations, we have \(F(z)=O((z-z_{0}))\), consequently, and so \(z_{0}\) is a zero of \(F(z)\).
Case 2. \(k\geq 2\). We have
and
Using the above two equalities, we get
By substituting the above equalities into (13) and performing some easy calculations, we again get \(F(z)=O((z-z_{0}))\). It also shows that \(z_{0}\) is a zero of \(F(z)\). □
Definition 2.1
([3])
Let f be a nonconstant meromorphic function in the complex plane and k be a positive integer. We call \(M[f]=f^{n_{0}}(f')^{n_{1}}\cdots (f^{(k)})^{n_{k}}\) a differential monomial in f, where \(n_{0},n_{1},\ldots,n_{k}\) are nonnegative integers, and \(\gamma _{M}:=n_{0}+n_{1}+\cdots +n_{k}\) its degree. Further, let \(M_{j}[f]\) denote differential monomials in f of degree \(\gamma _{M_{j}}\) for \(j=1,2,\ldots,k\), and let \(\alpha _{j}\) be meromorphic functions satisfying \(T(r,\alpha _{j})=S(r,f)\) for \(j=1,2,\ldots,k\), then \(P[f]=\alpha _{1}M_{1}[f]+\alpha _{2}M_{2}[f]+\cdots +\alpha _{k}M_{k}[f]\) is called a differential polynomial in f of degree \(\gamma _{P}:=\max_{1\leq j\leq k}\gamma _{M_{j}}\). If the coefficients \(\alpha _{j}\) only satisfy \(m(r,\alpha _{j})=S(r,f)\), then we call the function \(P[f]\) a quasi-differential polynomial in f.
Lemma 2.5
([3])
Let f be a nonconstant meromorphic function and \(Q^{*}[f]\), \(Q[f]\) be quasi-differential polynomials in f with \(Q[f]\not \equiv 0\). Let n be a positive integer and \(f^{n}Q^{*}[f]=Q[f]\). If \(\gamma _{Q}\leq n\), then \(m(r,Q^{*}[f])=S(r,f)\), where \(\gamma _{Q}\) is the degree of \(Q[f]\).
3 Proofs of theorems
In this section, we mainly give complete proofs for our main results.
Proof
of Theorem 1.6. In what follows, we consider two cases.
Case 1. When \(l\geq 3\), \(n\geq 1\), \(k\geq 1\), by inequality (4), we have
Case 2. When \(l=2\), \(n\geq 1\), \(k\geq 1\), we consider two subcases.
Subcase 2.1. First we suppose that \(k\geq 2\). From Lemma 2.3 and Lemma 2.4, we see immediately that \(F\not \equiv 0\) and simple poles of \(f(z)\) are the zeros of \(F(z)\). We also conclude that the poles of \(F(z)\) are with multiplicities two at most, which come from the multiple poles of \(f(z)\), or from the zeros of \(g(z)\), or from the zeros of \(h(z)\).
Set \(\beta =2f'(f^{(k)})^{n}+nf(f^{(k)})^{n-1}f^{(k+1)}-f(f^{(k)})^{n} \frac{g'}{g}\). Then \(f\beta =-a\frac{g'}{g}\) and \(h=-\frac{1}{a}\beta g\). We now consider the poles of \(\beta ^{2}F\). We note that the multiple poles of f with multiplicity \(q(\geq 2)\) are the zeros of β with multiplicity \(q-1\), and the zeros of h are either the zeros of g or the zeros of β. Thus,
since the poles of \(\beta ^{2}F\) come only from the zeros of g, and the multiplicity of poles of \(\beta ^{2}F\) is 4 at most.
Noting that \(m(r,F)=S(r,f)\) and \(m(r,\beta ^{2})=S(r,f)\) from Lemma 2.5, we have \(m(r,\beta ^{2}F)=S(r,f)\). Therefore,
Since the simple poles of f are the zeros of \(\beta ^{2}F\), hence
It follows from (7) and (26) that
i.e.
which leads to
We note that
and
By combining the above two inequalities and (27), we have
Subcase 2.2. Suppose that \(k=1\). Set \(\beta =2(f')^{n+1}+nf(f')^{n-1}f''-f(f')^{n}\frac{g'}{g}\). Then \(f\beta =-a\frac{g'}{g}\) and \(h=-\frac{1}{a}\beta g\). We again consider the poles of \(\beta ^{2}F\).
Arguing similarly as in Subcase 2.1, we have
and (26) is still valid.
It follows from (7) and (26) that
which yields
Noting that \(N(r,f)+N_{1}(r,f)-2\overline{N}(r,f)\geq 0\) and \(N (r,\frac{1}{f} )-\overline{N} (r,\frac{1}{f} )\geq 0\), we have
Thus, from the above two cases, we have
for \(M=\min \{\frac{1}{l-2},6\}\) and positive integers \(l(\geq 2)\), \(n(\geq 1)\), \(k(\geq 1)\). □
Proof of Corollary 1.1
Set \(\psi:=f^{l}(f^{(k)})^{n}\), where \(l(\geq 2)\), \(n(\geq 1)\), \(k(\geq 1)\) are integers. It follows from Lemma 2.1 that \(\psi \not \equiv 0\). By using Theorem 1.6, we have
where \(M=\min \{\frac{1}{l-2},6\}\).
Applying the lemma of the logarithmic derivative, we get
and
It follows from (31) and (32) that
Combining (30), (31), and (33), we have
where \(M=\min \{\frac{1}{l-2},6\}\). By the definition of the deficiency \(\Theta (a,\psi )\) and (33), we have, for \(M=\min \{\frac{1}{l-2},6\}\),
 □
Availability of data and materials
Not applicable.
References
Alotaibi, A.: On the zeros of \(af(f^{(k)})^{n}-1\) for \(n\geq 2\). Comput. Methods Funct. Theory 4(1), 227–235 (2004)
Bergweiler, W., Eremenko, A.: On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoam. 11(2), 355–373 (1995)
Doeringer, W.: Exceptional values of differential polynomials. Pac. J. Math. 98(1), 55–62 (1982)
Hayman, W.K.: Meromorphic Function. Clarendon Press, Oxford (1964)
Hayman, W.K.: Picard values of meromorphic functions and their derivatives. Ann. Math. 70(1), 9–42 (1959)
Huang, X.J., Gu, Y.X.: On the value distribution of \(f^{2}f^{(k)}\). J. Aust. Math. Soc. 78(1), 17–26 (2005)
Jiang, Y., Huang, B.: A note on the value distribution of \(f^{l}(f^{(k)})^{n}\). Hiroshima Math. J. 46(2), 135–147 (2016)
Karmakar, H., Sahoo, P.: On the value distribution of \(f^{n}f^{(k)}-1\). Results Math. 73(3), Article IDÂ 98 (2018)
Lahiri, I., Dewan, S.: Inequalities arising out of the value distribution of a differential monomial. J. Inequal. Pure Appl. Math. 4(2), Article IDÂ 27 (2003)
Laine, I.: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993)
Langley, J.K.: The zeros of \(ff''-b\). Results Math. 44(1–2), 130–140 (2003)
Mues, E.: Über ein Problem von Hayman. Math. Z. 164(3), 239–259 (1979)
Tse, C.K., Yang, C.C.: On the value distribution of \(f^{l}(f^{(k)})^{n}\). Kodai Math. J. 17(1), 163–169 (1994)
Wang, Y.F., Yang, C.C., Yang, L.: On the zeros of \(f(f^{(k)})^{n}-a\). Kexue Tongbao. 38, 2215–2218 (1993)
Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Science Press, Beijing (1995) Kluwer Academic, Dordrecht (2003)
Yang, L.: Value Distribution Theory. Springer, Berlin (1993)
Zhang, Q.D.: A growth theorem for meromorphic function. J. Chengdu Inst. Meteor. 20(1), 12–20 (1992)
Acknowledgements
The authors would like to thank the referee for his/her reading of the original version of the manuscript with valuable suggestions and comments.
Funding
The first author was supported by the NNSF of China (Nos: 12001117, 12001503), Basic and applied basic research of Guangzhou Basic Research Program (No. 202102020438). The second author was supported by the NNSF of China (Nos: 11801093, 11871260).
Author information
Authors and Affiliations
Contributions
All authors completed the main part of this article and corrected the main theorems. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, MF., Huang, ZB. Characteristic estimation of differential polynomials. J Inequal Appl 2021, 181 (2021). https://doi.org/10.1186/s13660-021-02716-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-021-02716-6