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Nonexistence of finite order solutions of certain second order linear differential equations. (English) Zbl 0879.34006
The author considers the second order differential equation \(f''+A(z)f'+ B(z)= 0\). \(A(z)\) and \(B(z)\) are entire functions not identically zero. \(\rho\) is the order of the entire function. This paper investigates the relationship of the order of the solution \(f\) to the order of the coefficients of the differential equation. The author proves these results:
Theorem I: If (1) \(A(z)\) is an entire function of finite nonintegral order \(\rho (A)>1\), and all its zeros lie in the angular sector \(\theta_1\leq \arg z\leq \theta_2\) satisfying \(\theta_2- \theta_1< {\pi \over q+1}\), if \(q\) is odd, and \(\theta_2- \theta_1< {\pi(2q-1) \over (q+1)2q}\) if \(q\) is even, \(q\) is the genus of \(A(z)\), (2) \(B(z)\) an entire function with \(0<\rho (B)<1/2\) then every nonconstant solution \(f\) of the differential equation has an infinite order satisfying \(\varlimsup_{r\to \infty} {\log\log T(r,f) \over \log r} \geq\rho (B)\).
The second result deals with the order of the solution of the differential equation when the coefficients are polynomials of equal degree over the complex numbers.
Reviewer: H.S.Nur (Fresno)

34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
30D20 Entire functions of one complex variable, general theory
Full Text: DOI
[1] P. O. BARRY, Some theorems related to the cos p theorem, Proc. London Math. Soc. (3), 21 (1970), 334-360. · Zbl 0204.42302 · doi:10.1112/plms/s3-21.2.334
[2] R. P. BOAS, Entire Functions, Academic Press, New York, 1954 · Zbl 0058.30201
[3] G. GUNDERSEN, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc., 305 (1988), 415-429 Zentralblatt MATH: · Zbl 0669.34010 · doi:10.2307/2001061 · www.zentralblatt-math.org
[4] G. GUNDERSEN, Estimates for the logarithmic derivative of a meromorphic func tion, plus similar estimates, J. London Math. Soc. (2), 37 (1988), 88-104. · Zbl 0638.30030 · doi:10.1112/jlms/s2-37.121.88
[5] W. HAYMAN, Meromorphic Functions, Clarendon Press, Oxford, 1964 · Zbl 0115.06203
[6] S. HELLERSTEIN, J. MILES AND J. Rossi, On the growth of solutions of /”+?/ + /=0, Trans. Amer. Math. Soc., 323 (1991), 693-706. · Zbl 0719.34011 · doi:10.2307/2001737
[7] K. KWON, On the growth of entire functions satisfying second order linear dif ferential equations, Bull. Korean Math. Soc. (2), 33 (1996), 487-496. · Zbl 0863.34007
[8] A. I. MARKUSHEVICH, Theory of Functions of a Complex Variable (Vol.11), Prentice-Hall, New Jersey, 1965 · Zbl 0142.32602
[9] M. OZAWA, On a solution of w^+e zw/+(az+b’)w=Q. Kodai Math. J., 3 (1980), 295-309 · Zbl 0463.34028 · doi:10.2996/kmj/1138036197
[10] G. VALIRON, Sur les functions entieres d’ordre fini et d’ordre nul, et en particu lier les functions a correspondance reguliere, Ann. Fac. Sci. Univ. Toulouse, 3 (1913), 117-257. · JFM 46.1462.03
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