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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)

MSC:
 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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References:
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