Functional inequalities, Jacobi products, and quasiconformal maps (Q1328072)

The authors give new identities and inequalities for the special function \[ \phi_ K(r)= \mu^{-1}(\mu(r)/K), \] where \(K\in (0,\infty)\), and \(r\in (0,1)\). The function \(\mu\) is defined as follows \[ \mu(r)= {\pi\over 2} {{\mathcal K}'(r)\over {\mathcal K}(r)}, \] where \[ {\mathcal K}(r)= \int^ 1_ 0 {dx\over \sqrt{(1- x^ 2)(1- r^ 2 x^ 2)}}, \] \({\mathcal K}'(r)= {\mathcal K}(r')\), and \(r'= \sqrt{(1- x^ 2)}\).