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Functional inequalities, Jacobi products, and quasiconformal maps. (English) Zbl 0799.30011
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 $$r'= \sqrt{(1- x^ 2)}$$.

##### MSC:
 30C62 Quasiconformal mappings in the complex plane 33E05 Elliptic functions and integrals