zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Growth of solutions of second order linear differential equations. (English) Zbl 1151.34069

The authors study the growth of solutions of the linear differential equation

f '' +A 1 (z)e az f ' +A 0 (z)e bz f=H(z),(2)

where A 1 (z), A 0 (z) and H(z) are entire functions of order less than one, and a,b.

The authors prove the following theorems

Theorem 1.1. Suppose that A 0 ¬0, A 1 ¬0, H are entire functions of order less than one, and the complex constants a, b satisfy ab0 and ab. Then every nontrivial solution f of (2) is of infinite order.

Theorem 1.3. Suppose that A 0 ¬0, A 1 ¬0, D 0 , D 1 , H are entire functions of order less than one, and the complex constants a, b satisfy ab0 and b/a<1. Then every nontrivial solution f of equation

f '' +(A 1 (z)e a·z +D 1 (z))f ' +(A 0 (z)e b·z +D 0 (z))f=H(z)

is of infinite order.

34M10Oscillation, growth of solutions (ODE in the complex domain)
[1]Barry, P. D.: On a theorem of Besicovitch, Quart. J. Math. Oxford ser. (2) 14, 293-302 (1963) · Zbl 0122.07602 · doi:10.1093/qmath/14.1.293
[2]Amemiya, I.; Ozawa, M.: Non-existence of finite order solutions of w&Prime;+e - zw ' +Q(z)w=0, Hokkaido math. J. 10, 1-17 (1981)
[3]Chen, Z. -X.: The growth of solutions of f&Prime;+e - zf ' +Q(z)f=0 where the order (Q)=1, Sci. China ser. A 45, 290-300 (2002)
[4]Frei, M.: Über die subnormalen lösungen der differentialgleichung w&Prime;+e - zw ' +(konst.)w=0, Comment. math. Helv. 36, 1-8 (1962) · Zbl 0115.06904 · doi:10.1007/BF02566887
[5]Fuchs, W.: Proof of a conjecture of G. Pólya concerning gap series, Illinois J. Math. 7, 661-667 (1963) · Zbl 0113.28702
[6]Gundersen, G.: On the question of whether f&Prime;+e - zf ' +B(z)f=0 can admit a solution f&nequiv;0 of finite order, Proc. roy. Soc. Edinburgh sect. A 102, 9-17 (1986)
[7]Gundersen, G.: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London math. Soc. (2) 37, 88-104 (1988) · Zbl 0638.30030 · doi:10.1112/jlms/s2-37.121.88
[8]Hayman, W.: Meromorphic functions, (1964) · Zbl 0115.06203
[9]Jank, G.; Volkmann, L.: Einführung in die theorie der ganzen und meromorphen funktionen mit anwendungen auf differentialgleichungen, (1985) · Zbl 0682.30001
[10]Kwon, K.; Kim, J.: Maximum modulus, characteristic, deficiency and growth of solutions of second order linear differential equations, Kodai math. J. 24, 344-351 (2001) · Zbl 1005.34079 · doi:10.2996/kmj/1106168809
[11]Laine, I.: Nevanlinna theory and complex differential equations, (1993)
[12]Laine, I.; Wu, P.: Growth of solutions of second order linear differential equations, Proc. amer. Math. soc. 128, 2693-2703 (2000) · Zbl 0952.34070 · doi:10.1090/S0002-9939-00-05350-8
[13]Langley, J.: On complex oscillation and a problem of ozawa, Kodai math. J. 9, 430-439 (1986) · Zbl 0609.34041 · doi:10.2996/kmj/1138037272
[14]Y. Li, J. Wang, Oscillation of solutions of linear differential equations, Acta Math. Sin. (Engl. Ser.), in press
[15]Markushevich, A.: Theory of functions of a complex variable, vol. II, (1985)
[16]Ozawa, M.: On a solution of w&Prime;+e - zw ' +(az+b)w=0, Kodai math. J. 3, 295-309 (1980)
[17]Yang, C. -C.; Yi, H. -X.: Uniqueness theory of meromorphic functions, (2003)