On the proximate order of growth of generating functions of Pólya frequency sequences (Q1851475)

The paper deals with the class of generating functions of Pólya frequency sequences of finite order \(r,r\in {\mathbb N},\) which is denoted by \(PF_r.\)( These sequences are also called \(r\)-multiply positive sequences by Fekete, Pólya, Schoenberg). The author studies the possible growth of analytic functions in the unit disk \(g\in PF_r, r>1.\) It is proved that for every integer \(r>1\) and every proximate order \(\rho (x) \to \rho, x\to \infty,\) where \(0<\rho <\infty,\) there exists a function \(g\in PF_r,\) analytic in \( {\bar {\mathbb C}}\setminus \{ 1\}\) and having an essential singularity of proximate order \(\rho (x)\) at \(z=1.\) Some classical function theory results concerning the connection between the order of growth of the analytic function in the unit disk and behavior of its Taylor coefficients are generalized to the case of proximate order. In the cases of \(\rho =0\) and \(\rho =\infty\) the examples of functions from \(PF_r,\) analytic in \( {\bar {\mathbb C}}\setminus \{ 1\}\) and having an essential singularity of order \(\rho\) at \(z=1\) are constructed for every integer \(r>1\).