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Removable sets in the oscillation theory of complex differential equations. (English) Zbl 0893.34004

The paper considers differential equations of the form \(f''+ A(z)f=0\), where \(A(z)\) is an entire function. The zero-distribution of the solution to this type of differential equation is considered from a geometric point of view whereby the author shows that the zero-distribution of \(E=f_1 f_2\) where \(f_1,f_2\) are two linearly independent solutions of the differential equation of uniform in a certain sense.
The author defines for \(s>r>1\), \(\varphi, \eta\) real numbers satisfying \(0\leq \varphi <2\pi\), \(0<\eta <\pi\) the annuli, \(D(r,s)= \{z=\rho e^{i\theta} \in\mathbb{C}: r<\rho <s\}\), and an annular rectangle, \(D(r,s, \varphi,\eta) =\{z= \rho e^{i \theta} \in\mathbb{C}: r<\rho<s, |\theta- \varphi |< \eta\}\).
Three theorems are then proven under the assumption that the exponents of convergence for the zero-sequences of the linearly independent solutions, \(f_1f_2\) satisfy \(\lambda (f_1) \lambda(f_2) =\infty\). It is then proven that the zero-sequences for \(E=f_1 f_2\) are uniformly distributed in the sense that quite large areas of the complex plane can be removed so that outside of these areas the contained zeros when counted for its exponents of convergence still remain infinite which regard to their maximums. A typical theorem proven is as follows:
Theorem 1. Given \(s>1\), let \((R_m)\) be a sequence of \(r\)-values such that \(R_{m+1} >sR_n\), and let \((\varphi_m)\) be a sequence of real numbers such that \(0\leq\varphi_m<2\pi\). Moreover, fix \(\eta\) and \(K\) so that \(0<\eta <\pi\) and \(1<K<s\), and consider \[ D:= \bigcup^\infty_{m=1} D(R_m, KR_m, \varphi_m, \eta). \] Let now \(A(z)\) be transcendental entire of finite order \(\rho (A)\), and let \(f_1, f_2\) be two linearly independent solutions of our differential equation such that for \(E:= f_1f_2\), \(\lambda(E) =\infty\). Then \(\lambda_{\mathbb{C} \setminus D} (E)= \infty\).
This paper is very well written.

MSC:

34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
34A26 Geometric methods in ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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References:

[1] Bank, S., On determining the location of complex zeros of certain linear differential equations, Ann. Mat. Pura Appl., 151, 67-96 (1988) · Zbl 0662.34005
[2] Bank, S.; Laine, I., On the oscillation theory of \(f Af A\), Trans. Amer. Math. Soc., 273, 351-363 (1982) · Zbl 0505.34026
[3] Bank, S.; Laine, I.; Langley, J., On the frequency of zeros of solutions of second-order linear differential equations, Resultate Math., 10, 8-24 (1986) · Zbl 0635.34007
[4] Chuang, C. T., Sur les fonctions-type, Sci. Sinica, 10, 171-181 (1961) · Zbl 0102.04704
[5] Gol’dberg, A. A.; Ostrovskii, I. V., The Distribution of Values of Meromorphic Functions (1970), Nauka: Nauka Moskow
[6] Gundersen, G., On the real zeros of solutions of \(fAzfAz\), Ann. Acad. Sci. Fenn. Ser. A I Math., 11, 275-294 (1986) · Zbl 0607.34007
[7] Hayman, W. K., Meromorphic Functions (1964), Clarendon: Clarendon Oxford · Zbl 0115.06203
[8] Laine, I., Nevanlinna Theory and Complex Differential Equations (1993), de Gruyter: de Gruyter Berlin
[9] Mokhon’ko, A. Z., An estimate of the modulus of the logarithmic derivative of a function which is meromorphic in an angular region, and its application, Ukrain. Mat. Zh., 41, 839-843 (1989) · Zbl 0686.30022
[10] Nevanlinna, R., Über die Eigenschaften meromorpher Funktionen in einem Winkelraum, Acta Soc. Sci. Fenn., 50, 1-45 (1925) · JFM 51.0257.02
[11] Rossi, J., Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc., 97, 61-66 (1986) · Zbl 0596.30047
[13] Wang, S. P., On the sectorial oscillation theory of \(fAzf\), Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 92 (1994) · Zbl 0824.30020
[14] Wu, S. J., Further results on Borel removable set of entire functions, Ann. Acad. Sci. Fenn. Ser. A I Math., 19, 67-81 (1994) · Zbl 0791.30023
[15] Wu, S. J., On the location of zeros of solutions of \(f Af Az\), Math. Scand., 74, 293-312 (1994) · Zbl 0827.34003
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