Canonical products of small order and related Pick functions (Q1765469)

The authors consider canonical products \[ P_\rho (z)=\prod_{n=1}^\infty(1+z/n^\rho),\quad\rho>1 \] and \[ Q(z)=\prod_{n=0}^\infty(1+zq^n),\quad q\in(0,1). \] It is shown that \((\log P_\rho(z))/z^{1/\rho}\) and \((\log Q(z)-\log Q(-q))/{\text{ Log}}\,((1+z)/(1-z))\) are Pick functions. Applications are found.