On Laurent series with multiply positive coefficients (Q2567424)

Denote by \(TQ_3 \) the class of all sequences \(\{ a_n \} _{n=-\infty} ^\infty ,\quad a_0\neq 0,\) such that all truncated sequences \(\{ a_k\} _{k=-n}^n:=\{ \dots ,0,0,a_{-n},a_{-n+1},\cdots ,a_n,0,0,\ldots \},\) are \(3-\)times positive in the sense of Pólya and Fekete for all \(n=1,2,\cdots\). If the sequence of coefficients of a Laurent series \(f(z)=\sum _{n=-\infty }^\infty a_nz^n\) belongs to the class \(TQ_3\), then we say that \(f\in TQ_3.\) The necessary and sufficient conditions under which a function \(f\) or a sequence \(\{ a_n\} \)belongs to the class \(TQ_3\) are given. Further, an estimate of the order of growth of functions in \(TQ_3\) is found.