On the hyper-order of solutions of two class of complex linear differential equations
- Wen Ping Huang^{1},
- Jing Lun Zhou^{1},
- Jin Tu^{2}Email author and
- Ju Hong Ning^{2}
DOI: 10.1186/s13662-015-0571-y
© Huang et al. 2015
Received: 28 February 2015
Accepted: 10 July 2015
Published: 29 July 2015
Abstract
We investigate the hyper-order of solutions of two class of complex linear differential equations. We investigate the growth of solutions of higher order and certain second order linear differential equations, and we obtain some results which improve and extend some previous results in complex oscillations.
Keywords
linear differential equation entire function hyper-order maximum modulusMSC
30D35 34M101 Introduction and results
Theorem A
(see [10])
Suppose that \(\rho(B)<\rho(A)<\infty\) and that \(T(r,A)\sim\log M(r,A)\) as \(r\rightarrow\infty\) outside a set of finite logarithmic measure. Then every nonconstant solution f of (1.1) is of infinite order.
Theorem B
(see [12])
Let \(A_{j}(z)\) (\(j=0,\ldots,k-1\)), \(F(z)\not\equiv0\) be entire functions. Suppose that there exists some \(d\in\{1,\ldots,k-1\}\) such that \(\max\{\rho(F), \rho(A_{j}): j\neq d\}=\rho<\rho(A_{d})<\infty\) and \(T(r,A_{d})\sim\log M(r,A_{d})\) as \(r\rightarrow\infty\) outside a set of upper logarithmic density less than \((\rho(A_{d})-\rho)/\rho(A_{d})\). Then every transcendental solution \(f(z)\) of (1.2) satisfies \(\overline{\lambda}(f)=\lambda(f)=\rho(f)=\infty\).
Theorem C
(see [12])
Let \(A_{j}(z)\) (\(j=0,\ldots,k-1\)), \(F(z)\equiv0\) be entire functions. Suppose that there exists some \(d\in\{1,\ldots,k-1\}\) such that \(\max\{\rho(A_{j}): j\neq0,d\}<\rho(A_{0})\leq\frac{1}{2}\) and that \(A_{d}(z)\) has a finite deficient value. Then every solution \(f(z)\not\equiv0\) of (1.2) satisfies \(\rho(A_{0})\leq\rho_{2}(f)\leq\rho(A_{d})\).
Then a natural question is: Can we estimate the hyper-order of the solutions of (1.1) and (1.2) under the same condition in Theorems A and B? And: Can we estimate the hyper-order of the solutions of (1.2) in Theorem C if \(\rho(A_{0})>\frac{1}{2}\)? Theorems 1.1 and 1.2 below give answers to the above questions.
Theorem D
(see [8])
- (i)If either \(\arg a_{n}\neq \arg b_{n}\) or \(a_{n}=cb_{n}\) with \(0< c<1\), then every nonconstant solution f ofhas infinite order with \(\rho_{2}(f)\geq n\).$$ f''+A_{1}(z)e^{P(z)}f'+A_{0}(z)e^{Q(z)}f=0 $$(1.4)
- (ii)
Let \(a_{n}=b_{n}\) and \(\operatorname{deg}(P-Q)=m\geq 1\), and let the orders of \(A_{1}(z)\) and \(A_{0}(z)\) be less than m. Then every nonconstant solution f of (1.4) has infinite order with \(\rho_{2}(f)\geq m\).
- (iii)
Let \(a_{n}=cb_{n}\) with \(c>1\) and \(\operatorname{deg}(P-cQ)=m\geq1\). Suppose \(\rho(A_{1})< m\) and \(0<\rho(A_{0})<\frac{1}{2}\). Then every nonconstant solution of (1.4) has infinite order with \(\rho_{2}(f)\geq\rho(A_{0})\).
- (iv)
Let \(a_{n}=cb_{n}\) with \(c>1\) and let \(P-cQ\) be a constant. Suppose that \(\rho(A_{1})<\rho(A_{0})<\frac{1}{2}\). Then every nonconstant solution of (1.4) has infinite order with \(\rho_{2}(f)\geq\rho(A_{0})\).
Theorem E
(see [13])
Theorem F
(see [14])
Theorem G
(see [15])
Theorem D left us a question: Can we have \(\rho_{2}(f)=n\) (n is a positive integer) for every nontrivial solution of (1.4) if \(a_{n}\neq b_{n}\)? Theorem E tells us that the question holds if \(n=1\). Many authors investigated the above question but none of them solve the question completely, and Theorem 1.3 completely solves this question. In the following, we give our results.
Theorem 1.1
- (i)
Every transcendental solution f of (1.2) satisfies \(\rho_{2}(f)=\rho(A_{d})\), and (1.2) may have polynomial solutions f of degree <d.
- (ii)
If \(F(z)\not\equiv0\), then every transcendental solution f of (1.2) satisfies \(\overline{\lambda}_{2}(f)=\lambda_{2}(f)=\rho_{2}(f)=\rho(A_{d})\).
- (iii)
If \(d=1\), then every nonconstant solution f of (1.2) satisfies \(\rho_{2}(f)=\rho(A_{1})\). Furthermore, if \(F(z)\not\equiv 0\), then every nonconstant solution f of (1.2) satisfies \(\overline{\lambda}_{2}(f)=\lambda_{2}(f)=\rho_{2}(f)=\rho(A_{1})\).
Theorem 1.2
Let \(A_{j}\) (\(j=0,\ldots,k-1\)), \(F(z)\equiv0\) be entire functions satisfying \(\max\{\rho(A_{j}): j\neq0,d\}<\rho(A_{0})<\rho(A_{d})<\infty\). Suppose that \(T(r,A_{0})\sim\log M(r,A_{0})\) as \(r\rightarrow\infty\) outside a set of r of finite logarithmic measure and that \(A_{d}\) has a finite deficient value. Then every solution \(f\not\equiv0\) of (1.2) satisfies \(\rho(A_{0})\leq\rho_{2}(f)\leq\rho (A_{d})\).
Theorem 1.3
2 Lemmas
Lemma 2.1
(see [16], p.399)
Let \(A_{j}(z)\) (\(j=0,\ldots,k-1\)), \(F(z)\) be entire functions satisfying \(\max\{\rho(A_{j}),\rho(F):j=0,\ldots,k-1\}\leq\rho<\infty\). Then every solution f of (1.2) satisfies \(\rho_{2}(f)\leq\rho\).
Lemma 2.2
(see [17])
- (i)There exist a constant \(B>0\) and a set \(E_{1}\subset(0,\infty)\) having finite logarithmic measure such that, for all z satisfying \(|z|=r\notin E_{1}\), we have$$ \biggl\vert \frac{f^{(j)}(z)}{f^{(i)}(z)}\biggr\vert \leq B\biggl[\frac{T(\alpha r,f)}{r}(\log r)^{\alpha}\log T(\alpha r,f)\biggr]^{j-i} \quad (0\leq i< j). $$(2.1)
- (ii)There exist a set \(H_{1}\subset[0,2\pi)\) that has linear measure zero and a constant \(B>0\) that depends only on α, for any \(\theta\in[0,2\pi)\backslash H_{1}\), there exists a constant \(R_{0}=R_{0}(\theta)>1\) such that, for all z satisfying \(\arg z=\theta\) and \(|z|=r>R_{0}\), we have$$ \biggl\vert \frac{f^{(j)}(z)}{f^{(i)}(z)}\biggr\vert \leq B\bigl[T(\alpha r,f)\log T( \alpha r,f)\bigr]^{j-i}\quad (0\leq i< j). $$(2.2)
Remark 2.1
We use \(E_{2}\subset(0,\infty)\) to denote a set of r of finite logarithmic measure throughout this paper, not necessarily the same at each occurrence.
Lemma 2.3
(see [18])
Lemma 2.4
(see [19])
Lemma 2.5
Proof
Remark 2.2
The following lemma is a special case of Lemma 29 in [12].
Lemma 2.6
(see [18])
Lemma 2.7
(see [20])
Lemma 2.8
(see [21])
Lemma 2.9
(see [22])
- (i)If \(\delta(P,\theta)>0\), then$$ \exp\bigl\{ (1-\varepsilon)\delta(P,\theta)r^{n}\bigr\} < \bigl\vert g \bigl(re^{i\theta}\bigr)\bigr\vert < \exp\bigl\{ (1+\varepsilon)\delta(P, \theta)r^{n}\bigr\} . $$(2.12)
- (ii)If \(\delta(P,\theta)<0\), then$$ \exp\bigl\{ (1+\varepsilon)\delta(P,\theta)r^{n}\bigr\} < \bigl\vert g \bigl(re^{i\theta}\bigr)\bigr\vert < \exp\bigl\{ (1-\varepsilon)\delta(P, \theta)r^{n}\bigr\} . $$(2.13)
3 Proofs of Theorems 1.1-1.3
Proof of Theorem 1.1
(ii) If \(F\not\equiv0\), by Lemma 2.8, we have that every transcendental solution \(f(z)\) of (1.2) satisfies \(\overline{\lambda}_{2}(f)=\lambda_{2}(f)=\rho_{2}(f)=\rho(A_{d})\).
(iii) If \(d=1\), it is easy to see that (1.2) cannot have polynomial solutions, and by (i) and (ii), we see that every nonconstant solution \(f(z)\) of (1.2) satisfies \(\rho_{2}(f)=\rho(A_{1})\) and \(\overline{\lambda}_{2}(f)=\lambda _{2}(f)=\rho_{2}(f)=\rho(A_{1})\) if \(F\not\equiv0\). □
Proof of Theorem 1.2
Proof of Theorem 1.3
(1) By Lemma 2.1, it is easy to see that every solution \(f\not\equiv0\) of (1.7) satisfies \(\rho _{2}(f)\leq n\) and that (1.7) has no polynomial solutions by the assumption. In the following, we need to show that every transcendental solution f of (1.7) satisfies \(\rho_{2}(f)\geq n\). We divide the proof into two parts: (i) \(\arg a_{n}\neq\arg b_{n}\) or \(a_{n}=cb_{n}\) (\(0< c<1\)), (ii) \(a_{n}=cb_{n}\) (\(c>1\)).
(i) \(\arg a_{n}\neq\arg b_{n}\) or \(a_{n}=cb_{n}\) (\(0< c<1\)). By Theorem D(i), every solution \(f\not\equiv0\) of (1.7) satisfies \(\rho_{2}(f)\geq n\).
Combining (i) and (ii), we have every solution \(f\not\equiv0\) of (1.7) satisfies \(\rho_{2}(f)= n\).
(2) By Lemma 2.1, it is easy to see that every solution f of (1.7) satisfies \(\rho_{2}(f)\leq n\). It is easy to know that (1.7) has no polynomial solutions by the assumption. In the following, we need to show that every transcendental solution f of (1.7) satisfies \(\rho_{2}(f)\geq n\). Since \(a_{n}=cb_{n}\) (\(c<0\)), then we have \(\{\theta:\delta(a_{n}z^{n},\theta)>0\}\cap \{\theta:\delta(b_{n}z^{n},\theta)>0\}=\emptyset\) and \(\{\theta:\delta(a_{n}z^{n},\theta)>0\}\cup \{\theta:\delta(b_{n}z^{n},\theta)>0\}\cup H_{3}=[0,2\pi)\), where \(H_{3}\subset[0,2\pi)\) is a set of linear measure zero. For each sufficiently large circle \(|z|=r\), we have if \(z_{r}=re^{i\theta_{r}}\) is a point satisfying \(|f(z_{r})|=M(r,f)\), for any given \(\delta_{4}>0\), set \(I=[\theta_{r}-\delta_{4},\theta_{r}+\delta_{4}]\), then we have either \(m(I\cap\{\theta:\delta(a_{n}z^{n},\theta)>0\})>0\) or \(m(I\cap\{\theta:\delta(a_{n}z^{n},\theta)<0\})>0\). We divide the proof into two cases: (i) \(m(I\cap\{\theta:\delta(a_{n}z^{n},\theta)>0\})>0\), (ii) \(m(I\cap\{\theta:\delta(a_{n}z^{n},\theta)<0\})>0\).
(ii) \(m(I\cap\{\theta:\delta(a_{n}z^{n},\theta)<0\})>0\). Replacing \(|A_{1}e^{P(z)}|\) with \(|A_{0}e^{Q(z)}|\) on the left side of (3.25) and by the same reasoning in case (i), we can obtain \(\rho_{2}(f)\geq n\) for every transcendental solution of (1.7).
Combining (i) and (ii), every solution f of (1.7) satisfies \(\rho_{2}(f)= n\). Since \(F(z)\not\equiv0\), by Lemma 2.8, every solution f of (1.7) satisfies \(\overline{\lambda}_{2}(f)=\lambda_{2}(f)=\rho_{2}(f)=n\). □
Declarations
Acknowledgements
This project is supported by the National Natural Science Foundation of China (11301232, 11301233, 11261024), the Natural Science Foundation of Jiangxi Province in China (20132BAB211002, 20151BAB201004) and the Foundation of Education Bureau of Jiangxi Province in China (GJJ14271, GJJ14272).
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
- Hayman, WK: Meromorphic Functions. Clarendon, Oxford (1964) Google Scholar
- Laine, I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993) View ArticleGoogle Scholar
- Yi, HX, Yang, CC: The Uniqueness Theory of Meromorphic Functions. Science Press, Beijing (1995) (in Chinese) Google Scholar
- Chen, ZX, Yang, CC: Some further results on the zeros and growth of entire solutions of second order differential equations. Kodai Math. J. 22, 273-285 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Gundersen, G: Finite order solutions of second order linear differential equations. Trans. Am. Math. Soc. 305, 415-429 (1988) MathSciNetView ArticleGoogle Scholar
- Hellerstein, S, Miles, J, Rossi, J: On the growth of solutions of \(f''+gf'+hf=0\). Trans. Am. Math. Soc. 324, 693-706 (1991) MathSciNetMATHGoogle Scholar
- Kwon, KH: On the growth of entire functions satisfying second order linear differential equations. Bull. Korean Math. Soc. 33(2), 487-496 (1996) MathSciNetMATHGoogle Scholar
- Kwon, KH: Nonexistence of finite order solutions of certain second order linear differential equations. Kodai Math. J. 19, 378-387 (1996) MathSciNetView ArticleMATHGoogle Scholar
- Kwon, KH, Kim, JH: Maximum modulus, characteristic, deficiency and growth of solution of second order linear differential equations. Kodai Math. J. 24, 344-351 (2001) MathSciNetView ArticleMATHGoogle Scholar
- Laine, I, Wu, PC: Growth of solutions of second order linear differential equations. Proc. Am. Math. Soc. 128, 2693-2703 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Wu, PC, Zhu, J: On the growth of solutions to the complex differential equation \(f''+A(z)f'+B(z)f=0\). Sci. China Ser. A 54(5), 939-947 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Tu, J, Deng, GT: Growth of solutions of a class of higher order linear differential equation. Complex Var. Elliptic Equ. 53(7), 623-631 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Chen, ZX: The growth of solutions of \(f''+ e^{-z}f'+Q(z)f=0\) where the order \((Q)=1\). Sci. China Ser. A 45(3), 290-300 (2002) MathSciNetMATHGoogle Scholar
- Wang, J, Laine, I: Growth of solutions of second order linear differential equations. J. Math. Anal. Appl. 342, 39-51 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Belaïdi, B: Growth and oscillation related to a second order linear differential equation. Math. Commun. 18(1), 171-184 (2013) MathSciNetMATHGoogle Scholar
- Kinnunen, L: Linear differential equations with solutions of finite iterated order. Southeast Asian Bull. Math. 22(4), 385-405 (1998) MathSciNetMATHGoogle Scholar
- Gundersen, G: Estimates for the logarithmic derivate of a meromorphic function, plus similar estimates. J. Lond. Math. Soc. 37, 88-104 (1988) MathSciNetMATHGoogle Scholar
- Tu, J, Xu, HY, Liu, HM, Liu, Y: Complex oscillation of higher order linear differential equations with coefficients being lacunary series of finite iterated order. Abstr. Appl. Anal. 2013, Article ID 634739 (2013) MathSciNetView ArticleGoogle Scholar
- Tu, J, Yi, CF: On the growth of solutions of a class of linear differential equations with coefficients having the same order. J. Math. Anal. Appl. 340(1), 487-497 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Fuchs, W: Proof of a conjecture of G. Pólya concerning gap series. Ill. J. Math. 7, 661-667 (1963) MATHGoogle Scholar
- Cao, TB, Chen, ZX, Zheng, XM, Tu, J: On the iterated order of meromorphic solutions of higher order linear differential equations. Ann. Differ. Equ. 21(2), 111-122 (2005) MathSciNetMATHGoogle Scholar
- Markushevich, AI: Theory of Functions of a Complex Variable, vol. II. Prentice Hall, Englewood Cliffs (1965) Google Scholar