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http://dx.doi.org/10.4134/CKMS.c180494

ON ZEROS AND GROWTH OF SOLUTIONS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS  

Kumar, Sanjay (Department of Mathematics Deen Dayal Upadhyaya College University of Delhi)
Saini, Manisha (Department of Mathematics University of Delhi)
Publication Information
Communications of the Korean Mathematical Society / v.35, no.1, 2020 , pp. 229-241 More about this Journal
Abstract
For a second order linear differential equation f" + A(z)f' + B(z)f = 0, with A(z) and B(z) being transcendental entire functions under some restrictions, we have established that all non-trivial solutions are of infinite order. In addition, we have proved that these solutions, with a condition, have exponent of convergence of zeros equal to infinity. Also, we have extended these results to higher order linear differential equations.
Keywords
Entire function; meromorphic function; order of growth; exponent of convergence; complex differential equation;
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1 I. Amemiya and M. Ozawa, Nonexistence of finite order solutions $w^{{\prime}{\prime}}$ + $e^{-z}w^{\prime}$ + Q(z)w = 0, Hokkaido Math. J. 10 (1981), Special Issue, 1-17.
2 S. B. Bank, I. Laine, and J. K. Langley, On the frequency of zeros of solutions of second order linear differential equations, Results Math. 10 (1986), no. 1-2, 8-24. https://doi.org/10.1007/BF03322360   DOI
3 P. D. Barry, On a theorem of Besicovitch, Quart. J. Math. Oxford Ser. (2) 14 (1963), 293-302. https://doi.org/10.1093/qmath/14.1.293   DOI
4 A. S. Besicovitch, On integral functions of order < 1, Math. Ann. 97 (1927), no. 1, 677-695. https://doi.org/10.1007/BF01447889   DOI
5 M. Frei, Uber die subnormalen Losungen der Differentialgleichung $w^{{\prime}{\prime}}+e^{-z}{\cdot}w^{\prime}+konst.{\cdot}w=0$, Comment. Math. Helv. 36 (1962), 1-8. https://doi.org/10.1007/BF02566887   DOI
6 G. G. Gundersen, On the question of whether $f^{{\prime}{\prime}}+e^{-z}f^{\prime}+B(z)f=0$ can admit a solution f $\not\equiv$ 0 of finite order, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 1-2, 9-17. https://doi.org/10.1017/S0308210500014451   DOI
7 G. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. Lond. Math. Soc. (2) 37 (1988), no. 1, 88-104. https://doi.org/10. 1112/jlms/s2-37.121.88   DOI
8 G. G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), no. 1, 415-429. https://doi.org/10.2307/2001061   DOI
9 G. G. Gundersen, Research questions on meromorphic functions and complex differential equations, Comput. Methods Funct. Theory 17 (2017), no. 2, 195-209. https://doi.org/10.1007/s40315-016-0178-7   DOI
10 W. K. Hayman and J. F. Rossi, Characteristic, maximum modulus and value distribution, Trans. Amer. Math. Soc. 284 (1984), no. 2, 651-664. https://doi.org/10.2307/1999100   DOI
11 J. Heittokangas, I. Laine, K. Tohge, and Z. Wen, Completely regular growth solutions of second order complex linear differential equations, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 2, 985-1003. https://doi.org/10.5186/aasfm.2015.4057   DOI
12 S. Hellerstein, J. Miles, and J. Rossi, On the growth of solutions of $f^{{\prime}{\prime}}+gf^{\prime}+hf$, Trans. Amer. Math. Soc. 324 (1991), no. 2, 693-706. https://doi.org/10.2307/2001737   DOI
13 E. Hille, Lectures on Ordinary Differential Equations, A Wiley-Interscience Publication (JOHN WILEY & SONS), London, 1969.
14 A. S. B. Holland, Theory of Entire Function, Academic Press, New York, 1973.
15 I. Laine, Nevanlinna Theory and Complex Differential Equations, De Gruyter Studies in Mathematics, 15, Walter de Gruyter & Co., Berlin, 1993. https://doi.org/10.1515/9783110863147
16 J. K. Langley, On complex oscillation and a problem of Ozawa, Kodai Math. J. 9 (1986), no. 3, 430-439.   DOI
17 J. Long, Growth of solutions of second order complex linear differential equations with entire coefficients, Filomat 32 (2018), no. 1, 275-284. https://doi.org/10.2298/fil1801275l   DOI
18 J. Long, L. Shi, X. Wu, and S. Zhang, On a question of Gundersen concerning the growth of solutions of linear differential equations, Ann. Acad. Sci. Fenn. Math. 43 (2018), no. 1, 337-348. https://doi.org/10.5186/aasfm.2018.4315   DOI
19 J. R. Long, P. C. Wu, and Z. Zhang, On the growth of solutions of second order linear differential equations with extremal coefficients, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 2, 365-372. https://doi.org/10.1007/s10114-012-0648-4   DOI
20 M. Ozawa, On a solution of $w^{{\prime}{\prime}}+e^{-z}w^{\prime}+(az+b)w=0$, Kodai Math. J. 3 (1980), no. 2, 295-309. http://projecteuclid.org/euclid.kmj/1138036197   DOI
21 X. Wu, J. R. Long, J. Heittokangas, and K. E. Qiu, Second-order complex linear differential equations with special functions or extremal functions as coefficients, Electron. J. Differential Equations 2015 (2015), No. 143, 15 pp.
22 S.-Z. Wu and X.-M. Zheng, On meromorphic solutions of some linear differential equations with entire coefficients being Fabry gap series, Adv. Difference Equ. 2015 (2015), 32, 13 pp. https://doi.org/10.1186/s13662-015-0380-3
23 L. Yang, Value Distribution Theory, translated and revised from the 1982 Chinese original, Springer-Verlag, Berlin, 1993.