## Abstract and Applied Analysis

### On the Growth of Solutions of a Class of Higher Order Linear Differential Equations with Extremal Coefficients

#### Abstract

We consider that the linear differential equations ${f}^{(k)}+{A}_{k-\mathrm{1}}(z){f}^{(k-\mathrm{1})}+\cdots +{A}_{\mathrm{1}}(z){f}^{\mathrm{\prime }}+{A}_{\mathrm{0}}(z)f=\mathrm{0}$, where ${A}_{j} (j=\mathrm{0,1},\dots ,k-\mathrm{1})$, are entire functions. Assume that there exists $l\in \{\mathrm{1,2},\dots ,k-\mathrm{1}\}$, such that ${A}_{l}$ is extremal for Yang's inequality; then we will give some conditions on other coefficients which can guarantee that every solution $f(\not\equiv \mathrm{0})$ of the equation is of infinite order. More specifically, we estimate the lower bound of hyperorder of $f$ if every solution $f(\not\equiv \mathrm{0})$ of the equation is of infinite order.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 305710, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412276900

Digital Object Identifier
doi:10.1155/2014/305710

Mathematical Reviews number (MathSciNet)
MR3198173

Zentralblatt MATH identifier
07022131

#### Citation

Long, Jianren; Qiu, Chunhui; Wu, Pengcheng. On the Growth of Solutions of a Class of Higher Order Linear Differential Equations with Extremal Coefficients. Abstr. Appl. Anal. 2014 (2014), Article ID 305710, 7 pages. doi:10.1155/2014/305710. https://projecteuclid.org/euclid.aaa/1412276900

#### References

• W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.
• I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of de Gruyter Studies in Mathematics, Walter de Gruyter, New York, NY, USA, 1993.
• L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993.
• C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557, Kluwer Academic Publishers, New York, NY, USA, 2003.
• G. G. Gundersen, “Finite order solutions of second order linear differential equations,” Transactions of the American Mathematical Society, vol. 305, no. 1, pp. 415–429, 1988.
• S. Hellerstein, J. Miles, and J. Rossi, “On the growth of solutions of ${f}^{{''}}+g{f}^{\prime }+hf=0$,” Transactions of the American Mathematical Society, vol. 324, no. 2, pp. 693–706, 1991.
• S. B. Bank and I. Laine, “On the oscillation theory of ${f}^{{''}}+Af=0$ where \emphA is entire,” Transactions of the American Mathematical Society, vol. 273, no. 1, pp. 351–363, 1982.
• Z.-X. Chen and C.-C. Yang, “Some further results on the zeros and growths of entire solutions of second order linear differential equations,” Kodai Mathematical Journal, vol. 22, no. 2, pp. 273–285, 1999.
• G. G. Gundersen, “On the question of whether ${f}^{{''}}+{e}^{-z}{f}^{\prime }+B(z)f=0$ can admit a solution $f\not\equiv0$ of finite order,” Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 102, no. 1-2, pp. 9–17, 1986.
• S. Hellerstein, J. Miles, and J. Rossi, “On the growth of solutions of certain linear differential equations,” Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, vol. 17, no. 2, pp. 343–365, 1992.
• K.-H. Kwon and J.-H. Kim, “Maximum modulus, characteristic, deficiency and growth of solutions of second order linear differential equations,” Kodai Mathematical Journal, vol. 24, no. 3, pp. 344–351, 2001.
• M. Ozawa, “On a solution of ${w}^{{''}}+{e}^{-z}{w}^{\prime }+(az+b)w=0$,” Kodai Mathematical Journal, vol. 3, no. 2, pp. 295–309, 1980.
• Z.-X. Chen and C.-C. Yang, “On the zeros and hyper-order of meromorphic solutions of linear differential equations,” Annales Academiæ Scientiarum Fennicæ. Mathematica, vol. 24, no. 1, pp. 215–224, 1999.
• K.-H. Kwon, “On the growth of entire functions satisfying second order linear differential equations,” Bulletin of the Korean Mathematical Society, vol. 33, no. 3, pp. 487–496, 1996.
• K.-H. Kwon, “Nonexistence of finite order solutions of certain second order linear differential equations,” Kodai Mathematical Journal, vol. 19, no. 3, pp. 378–387, 1996.
• L. Yang, “Deficient values and angular distribution of entire functions,” Transactions of the American Mathematical Society, vol. 308, no. 2, pp. 583–601, 1988.
• S. Wu, “Some results on entire functions of finite lower order,” Acta Mathematica Sinica. New Series, vol. 10, no. 2, pp. 168–178, 1994.
• J. R. Long, P. C. Wu, and Z. Zhang, “On the growth of solutions of second order linear differential equations with extremal coefficients,” Acta Mathematica Sinica, vol. 29, no. 2, pp. 365–372, 2013.
• B. Belaïdi, “Growth and oscillation theory of non-homogeneous complex differential equations with entire coefficients,” Communications in Mathematical Analysis, vol. 5, no. 2, pp. 13–25, 2008.
• B. Belaïdi and S. Abbas, “On the hyper order of solutions of a class of higher order linear differential equations,” Analele stiintifice ale Universitatii Ovidius Constanta, vol. 16, no. 2, pp. 15–30, 2008.
• Z. Chen, “On the hyper order of solutions of higher order differential equations,” Chinese Annals of Mathematics B, vol. 24, no. 4, pp. 501–508, 2003.
• Z. Chen, “On the growth of sulutions of a class of higher order differential equations,” Acta Mathematica Scientia, vol. 24, pp. 52–60, 2004.
• J. Tu and C.-F. Yi, “On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 487–497, 2008.
• G. G. Gundersen, “Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates,” Journal of the London Mathematical Society. Second Series, vol. 37, no. 1, pp. 88–104, 1988.
• P. D. Barry, “Some theorems related to the $\pi \rho$ theorem,” Proceedings of the London Mathematical Society, vol. 21, pp. 334–360, 1970.
• S. B. Bank, “A general theorem concerning the growth of solutions of first-order algebraic differential equations,” Compositio Mathematica, vol. 25, pp. 61–70, 1972.
• L. Yang and G. H. Zhang, “Distribution of Borel directions of entire functions,” Acta Mathematica Sinica, vol. 19, no. 3, pp. 157–168, 1976 (Chinese).
• M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, Japan, 1959.
• P. D. Barry, “On a theorem of Besicovitch,” The Quarterly Journal of Mathematics, vol. 14, pp. 293–302, 1963. \endinput