On the zeros of certain linear differential polynomials
Let f be a meromorphic function in the plane and let F be given by \[ F=f^{(k)}+a_{k-1}f^{(k-1)}+...+a_ 0f,\quad k\geq 3. \] The authors prove that there is no function f such that fF has no zeros, where \(a_ 1,...,a_{k-1}\) are constants, and \(a_ 0\) is a polynomial of degree one. In the case where the \(a_ j\) are polynomials they classify all functions f such that ff'F has no zeros. Here f has one of the following forms: \[ f=\exp (\int e^ Q+c),\quad f=c(\int e^ Q-b)^{-n}, \] \[ f(z)=(az+b)^{-n},\quad f(z)=\exp (az+b), \] where a,b,c are constants, Q is a polynomial, and n is a positive integer.
scientific article; zbMATH DE number 4171304