Three results in the value-distribution theory of solutions of linear differential equations
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.
scientific article; zbMATH DE number 4034141