- Research
- Open Access
Meromorphic solutions to certain class of differential equations in an angular domain
- Fengrong Zhang^{1}Email author,
- Hui Xu^{1},
- Mengmeng Zhang^{1} and
- Daiwei Liang^{1}
https://doi.org/10.1186/s13662-019-1993-8
© The Author(s) 2019
- Received: 7 September 2018
- Accepted: 27 January 2019
- Published: 5 March 2019
Abstract
In this paper, we study the admissible meromorphic solutions to the algebraic differential equation \(f^{n} f' + P_{n-1}( f ) = u\mathrm {e}^{v}\) in an angular domain instead of the whole complex plane, where \(P_{n-1}(f)\) is a differential polynomial in f of degree \(\leq n-1\) with small function coefficients, u is a non-vanishing small function of f and v is an entire function. Herein, mainly, we are able to show that the equation does not admit any meromorphic solution f under some conditions unless \(P_{n-1}(f)\equiv0\). Using this result, we are able to extend or generalize a well-known result of Hayman.
Keywords
- Nevanlinna theory on an angular domain
- Admissible solution
- Entire function
- Differential equation
MSC
- 30D35
- 30D20
- 30D30
1 Introduction
Theorem 1.1
([1])
It becomes natural for us to ask the following question: Does the conclusion of Theorem 1.1 remains valid if the condition “u is a non-zero rational function and v is a non-constant polynomial” is replaced by a weaker one, such as “u and v are entire functions” in the above theorem?
In this paper, we consider a slightly more general case of Eq. (1.1), where u and the coefficients of \(Q_{d}(z, f )\) are meromorphic functions, not necessarily rational functions. Furthermore, v is an entire function. In this case, we will show that there exists an angular domain \(\varOmega(\alpha,\beta)=\{z: \alpha\leq \arg z\leq\beta\}\) such that the conclusion of Theorem 1.1 remains valid, where \(\alpha,\beta\in[0,2\pi]\) and \(0< \beta-\alpha\leq2\pi\).
The technique developed here will be different from what has been employed in [1]. Now, we first introduce the Nevanlinna theory on an angular domain, which can be found in Goldberg–Ostrovskii [2] and the references therein.
Similarly, for any finite value a, we define here \(A_{\alpha\beta }(r,f_{a})\), \(B_{\alpha\beta}(r,f_{a})\), \(C_{\alpha\beta}(r,f_{a})\) and \(S_{\alpha\beta}(r,f_{a})\), where \(f_{a}=1/(f-a)\). In what follows, when there is no danger of confusion, next we omit the subscript of all the above notations and, respectively, use the notations \(A(r,f_{a})\), \(B(r,f_{a})\), \(C(r,f_{a})\) and \(S(r,f_{a})\) instead of \(A_{\alpha\beta}(r,f_{a})\), \(B_{\alpha\beta}(r,f_{a})\), \(C_{\alpha\beta}(r,f_{a})\) and \(S_{\alpha\beta}(r,f_{a})\) for any finite value a.
In this paper, unless otherwise stated, we address a meromorphic function that is defined and meromorphic in the whole complex plane. It is assumed the reader is familiar with the basic theory of the Nevanlinna value distribution and its standard symbols and notations (see e.g., [4, 5]).
For the order \(\rho(f)=\infty\), we use the following concept of a proximate order as introduced in [6].
Theorem 1.2
([6])
- (i)
\(\rho(r)\) is continuous and nondecreasing for \(r>r_{0}\) (\(r_{0}>0\))and tends to +∞ as \(r\to+\infty\).
- (ii)The function \(U(r)=r^{\rho(r)}\) (\(r\geq r_{0}\)) satisfies the condition$$\lim_{r\to+\infty}\frac{\log U(R)}{ \log U(r)}=1,\quad R=r+\frac{ r}{\log U(r)}. $$
- (iii)$$\limsup_{r\to+\infty}\frac{\log B(r)}{\log U(r)}=1. $$
For the sake of simplicity, we use the Landau symbols \(O(\cdots)\) and \(o(\cdots)\) as \(r\to\infty\), where and in what follows, \(Q(r,f)\) is such a quantity that if the order \(\rho(f) < \infty\), then \(Q(r,f)=O(1)\), as \(r\to\infty\); if the order \(\rho(f)=\infty\), then \(Q(r,f)=O(\log U(r))\). It is not necessarily the same for every occurrence in the context.
In addition, we need the following concept (see, e.g., [7] and [8]).
Definition 1.3
Let \(R(z,\omega)\) be rational in ω with meromorphic coefficients. A meromorphic solution ω of \((\omega ')^{n}=R(z,\omega)\) is called admissible, if \(S(r,a)=Q(r,\omega)\) holds for all coefficients a of \(R(z,\omega)\).
Of course, admissibility makes sense relative to any family of meromorphic functions, without any reference to differential equations.
Before proceeding further, we recall the following result. Recently, Liu–Lü–Yang considered the possible admissible solution to the following equation.
Theorem 1.4
([9])
Motivated by the preceding, as a continuation and further studies on some of the related results in the complex plane, we will state our main results in Sect. 3, which extend some results earlier; see, e.g., [10–13] and the references therein.
2 Preliminary lemmas
To prove our results, the following lemmas are needed.
Lemma 2.1
([2, pp. 23–26] and [2, Theorem 3.1])
Lemma 2.2
([2, Theorem 3.3])
Lemma 2.3
([2])
Lemma 2.4
We omit the proof of Lemma 2.4, since it is similar to the proof of Clunie’s theorem [4, p. 68, Lemma 3.3] and [14, 15] in the plane.
3 Main results
In this section, we give our main results as follows.
Theorem 3.1
Proof
Two cases will now be considered below, depending on whether or not \((2n+1)\varphi f''(z)-(t\varphi+n\varphi')f'(z)\) vanishes identically.
This completes the proof of our conclusion, namely \(f^{n}f'+P_{n-1}(f)=u\mathrm{e}^{v}\) does not admit any meromorphic solution f on \(\varOmega(\theta-\varepsilon,\theta+\varepsilon)\) with \(C(r,f)=Q(r,f)\) unless \(P_{n-1}(f)\equiv0\).
Corollary 3.2
Let f be a transcendental meromorphic function, \(P_{n-1}(f)\) (\(P_{n-1}(0)\neq0\)) denote a differential polynomial in f with its coefficients are in \(S_{f}\) and \(\deg P_{n-1}(f)\leq n-1\). Then, for any positive integer n, and any ε (\(0<\varepsilon <\pi/2\)), there exists a direction \(\arg z=\theta\) such that \(f^{n}f'+P_{n-1}(f)\) has infinitely many zeros on \(\varOmega(\theta -\varepsilon,\theta+\varepsilon)\) with \(C(r, f ) =Q(r, f )\).
Remark 3.3
Take \(f(z)=\mathrm{e}^{z}+\mathrm{e}^{-z}\), and \(P_{0}(f)=-2\). Obviously, \(ff''+P_{0}(f)=\mathrm{e}^{2z}+\mathrm{e}^{-2z}\) has infinitely many zeros.
Based on Corollary 3.2 and Remark 3.3, we present the following more general conjecture; see, e.g., [3] and [17].
Conjecture 3.4
Remark 3.5
We will give some examples below to show that the restricted conditions in Corollary 3.2 are sharp.
Example 3.6
Take \(f(z)=\mathrm{e}^{\mathrm{e}^{z}}-1\), and \(P_{2}(f)=2\mathrm {e}^{z}f^{2}+3\mathrm{e}^{z}f+\mathrm{e}^{z}\). Obviously, \(\mathrm{e}^{z}\in S_{f}\), and \(f^{2}f'+P_{2}(f)=\mathrm{e}^{z}\mathrm{e}^{3\mathrm{e}^{z}}\) has no zeros on \(\varOmega(\theta-\varepsilon,\theta+\varepsilon)\).
Example 3.7
Let \(f(z)=\mathrm{e}^{z^{2}}-1\), and \(P_{1}(f)=2zf+2z\). Obviously, \(ff'+P_{1}(f)=2z\mathrm{e}^{2z^{2}}\) has only one zero on \(\varOmega(\theta-\varepsilon,\theta+\varepsilon)\).
Examples 3.6 and 3.7 tell us that the conclusion of Corollary 3.2 is false, if we replace the restricted condition “\(\deg P_{n-1}(f)\leq n-1\)” by “\(\deg P_{n}(f)\leq n\)”.
Example 3.8
If we take \(f(z)=z^{2}\mathrm{e}^{z}\), \(P_{2}(f)=ff'-(\frac{2}{z}+1)f^{2}\), then \(2/z+1\in S_{f}\), and \(f^{3}f'+P_{2}(f)=(2+z)z^{7}\mathrm{e}^{4z}\) has only finitely many zeros on \(\varOmega(\theta-\varepsilon,\theta +\varepsilon)\).
Example 3.8 shows that the restricted condition “\(P_{n-1}(0)\neq 0\)” in Corollary 3.2 is necessary.
Corollary 3.2 may be false if the condition “\(C(r,f)=Q(r,f)\)” is violated. There is no difficulty in showing that Example 3.9 below is a counterexample.
Example 3.9
We take \(\alpha= -\pi\), \(\beta= \pi\), \(f(z)=\frac{\mathrm{e}^{z}}{\mathrm {e}^{z}-1}\), \(P_{1}(f)=\frac{3}{2}f''+\frac{3}{2}f'+f-1\).
Indeed, a calculation yields \(C(r,1/f)=0\), \(C(r,1/(f-1))=0\). It follows by Lemma 2.2 that \(C(r,f)=S(r,f)+Q(r,f)\). Further, we would like to mention that \(f^{2}f'+P_{1}(f)=-\frac{1}{(\mathrm{e}^{z}-1)^{4}}\) has no zeros on \(\varOmega(\theta-\varepsilon,\theta+\varepsilon)\).
From Theorem 3.1, we also obtain the following result, which complements the corresponding results in [1].
Theorem 3.10
Let \(f\in M(\rho(r))\) of finite order with \(C(r,f)=Q(r,f)\), \(q_{m}( f ) = b_{m} f^{m} +\cdots+ b_{1} f + b_{0}\) a polynomial of degree m with \(b_{j}\in S_{f}\) (\(j=0,1,\ldots,m\)), and let n be an integer with \(n\geq m + 1\). Then, for any ε (\(0<\varepsilon<\pi/2\)), there exists a direction \(\arg z=\theta\) such that \(f' f^{n} + q_{m}( f )\) assumes every function \(\gamma\in S_{f}\) infinitely many times on \(\varOmega (\theta-\varepsilon,\theta+\varepsilon)\), except for a possible function \(b_{0} = q_{m}(0)\). On the other hand, if \(f' f^{n} + q_{m}( f )\) assumes \(b_{0} = q_{m}(0)\) finitely many times on \(\varOmega(\theta-\varepsilon,\theta +\varepsilon)\), then \(q_{m}(z)\equiv b_{0}\).
Proof
This theorem can be proved in the same manner as that in the proof of Theorem 3.1, so it is omitted here. □
4 Conclusions
Using the different and much simpler proofs, this paper provides two main results on \(\varOmega(\alpha,\beta)\), which extend the main results that were derived in [1]. To bring about our results from the more general hypotheses without complicated calculations will probably be the most interesting feature of this note. And then some examples show that the restrict conditions are necessary. Finally, one more general conjecture was posed in this note.
Declarations
Acknowledgements
The authors would like to thank the referee for his/her several important suggestions and for pointing out some errors in our original manuscript. Professor Weiran Lü also gave some suggestions as regards the manuscript.
Funding
This work was supported by the Fundamental Research Funds for the Central Universities (No. 18CX02045A) and the National Natural Science Foundation of China (Nos. 11602305, 11601521).
Authors’ contributions
All authors contributed in drafting this manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Liao, L.W., Ye, Z.: On solutions to nonhomogeneous algebraic differential equations and their application. J. Aust. Math. Soc. 97, 391–403 (2014) MathSciNetView ArticleGoogle Scholar
- Goldberg, A.A., Ostrovskii, I.V.: Value Distribution of Meromorphic Functions. Translations of Mathematical Monographs, vol. 236. Am. Math. Soc., Providence (2008) Translated from the 1970 Russian original by Mikhail Ostrovskii MATHGoogle Scholar
- Yang, L., Yang, C.C.: Angular distribution of values of \(ff'\). Sci. China Ser. A 37, 284–294 (1994) MathSciNetMATHGoogle Scholar
- Hayman, W.K.: Meromorphic Functions. Clarendon, Oxford (1964) MATHGoogle Scholar
- Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Science Press, Beijing (2003) View ArticleGoogle Scholar
- Chuang, C.T.: Differential Polynomials of Meromorphic Functions. Beijing Normal University Press, Beijing (1999) (in English) Google Scholar
- Ishizaki, K., Yanagihara, N.: On admissible solutions of algebraic differential equations. Funkc. Ekvacioj 38, 433–442 (1995) MathSciNetMATHGoogle Scholar
- Laine, I.: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993) View ArticleGoogle Scholar
- Liu, N.N., Lü, W.R., Yang, C.C.: On the meromorphic solutions of certain class of nonlinear differential equations. J. Inequal. Appl. 2015, 149 (2015) MathSciNetView ArticleGoogle Scholar
- Langley, J.K.: On the zeros of \(ff''-b\). Results Math. 44, 130–140 (2003) MathSciNetView ArticleGoogle Scholar
- Lü, W.R., Liu, N.N., Yang, C.C., Zhuo, C.P.: Notes on the value distribution of \(ff^{(k)}-b\). Kodai Math. J. 39, 500–509 (2016) MathSciNetView ArticleGoogle Scholar
- Mues, E.: Über ein Problem von Hayman. Math. Z. 164, 239–259 (1979) MathSciNetView ArticleGoogle Scholar
- Zhang, J., Liao, L.: Admissible meromorphic solutions of algebraic differential equations. J. Math. Anal. Appl. 397, 225–232 (2013) MathSciNetView ArticleGoogle Scholar
- Clunie, J.: On integral and meromorphic functions. J. Lond. Math. Soc. 37, 17–27 (1962) MathSciNetView ArticleGoogle Scholar
- Yang, C.C., Ye, Z.: Estimates of the proximity function of differential polynomials. Proc. Jpn. Acad., Ser. A, Math. Sci. 83, 50–55 (2007) View ArticleGoogle Scholar
- Zheng, J.H.: Value Distribution of Meromorphic Functions. Tsinghua University Press, Beijing; Springer, Berlin (2010) MATHGoogle Scholar
- Yang, C.C., Yang, L., Wang, Y.F.: On the zeros of \((f^{(k)})^{n}f-a\). Chin. Sci. Bull. 38, 2125–2128 (1993) Google Scholar