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)
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