On the complex oscillation of some linear differential equations (Q678675)

The paper treats the linear differential equation, \[ f^{(k)}+ A(z)f=0,\tag{\(*\)} \] where \(k\geq 2\) is an integer, \(A(z)\) is a transcendental entire function of order \(\sigma(A)\). It is shown that any non-trivial solution of \((*)\) satisfies \(\lambda(f)\geq \sigma(A)\), where \(\lambda(f)\) is the exponent of convergence of the zero-sequence of \(f\), under the condition, \[ K\overline N\Biggl(r,{1\over A}\Biggr)\leq T(r,A),\quad r\neq E \] for \(K>2k\) and an exceptional set, \(E\), of finite linear measure. Herein \(N(r,f)\) and \(T(r,f)\) denotes the counting function and the characteristic function of \(f\) respectively. A very nice example is given demonstrating the result. Several technical lemmas are extremely well done preparing the proof of the theorem. The other half of the paper treats the second-order equation, \[ f''+ (e^{p_1(z)}+ e^{p_2(z)}+ Q(z)) f=0,\tag{\(**\)} \] where \(p_1(z)\) and \(p_2(z)\) are non-constant polynomials of degree \(n\) and \(m\) respectively. \(Q(z)\) is an entire function of order less than \(\max(m,n)\). Several theorems are proved regarding equation \((**)\), where once again several well done lemmas prepare the proof of the theorem. The paper is very well written.