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Three results in the value-distribution theory of solutions of linear differential equations. (English) Zbl 0635.34009
The author develops, through three results, new information about the distribution of zeros of all solutions $$f\not\equiv 0$$ for three different classes of equations of the form: (1) $$f''+A(z)f=0$$; $$A(z)=arbitrary$$ entire function, and thereby obtains a complete value- distribution theory for all solutions. The first result concerns a class of equations (1), where the order of A(z) can be any nonnegative real number, and shows that, if the equation possesses a solution $$f_ 1\not\equiv 0$$ satisfying $${\bar \lambda}$$(f$${}_ 1)<\infty$$, and which is of a certain form, then for any solution $$f_ 2$$, which is linearly independent with $$f_ 1$$, it must be valid $${\bar \lambda}$$(f$${}_ 2)=\infty$$. The second result concerns the value-distribution theory for the solutions of a class of equations (1) where A(z) is a periodic entire function of the form $$B(e^{az})$$ $$(B(\zeta)=rational$$ function; $$a=nonzero$$ constant). Firstly, it is proved that, the conclusion $${\bar \lambda}$$(f)$$=\infty$$ for every solution $$f\not\equiv 0$$, can be replaced by the stronger conclusion $$(2)\quad \log^+\bar N(r,1/f)\neq o(r)$$ as $$r\to +\infty$$. Furthermore it is shown that, instead of requiring both of the poles of B($$\zeta)$$ at $$\zeta =0$$ and $$\zeta =\infty$$ to be of odd order, the stronger conclusion (2) will hold for all solutions $$f\not\equiv 0$$, when at least one of these poles is of odd order. The third result is an improvement of an existing one in the literature result, namely of the value-distribution theory given for the solution of (1) in the case when A(z) is a nonconstant polynomial of degree n. The paper is original, with strict mathematical construction and it is very clearly written.
Reviewer: D.E.Panayotounakos

##### MSC:
 34A30 Linear ordinary differential equations and systems, general 34M99 Ordinary differential equations in the complex domain
##### Keywords:
distribution of zeros; value-distribution theory
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##### References:
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