Singular values, quasiconformal maps and the Schottky upper bound (Q1286713)

The author highlights the connection between Ramanujan's modular equation and certain aspects of quasiconformal mappings [see \textit{O. Lehto} and \textit{K. L. Virtanen}, Quasiconformal mappings in the plane (1973; Zbl 0267.30016)]. Some properties including asymptotically sharp bounds for singular values of Ramanujan's generalised modular equation are obtained. Infinite product representations of Hersch-Pfluger \(\phi\)-distortion function and the Agard \(\eta\)--distortion function are obtained. Thus sharp estimates for these special functions are obtained which are then used to improve related quasiconformal distortion theorems [cf. \textit{S. L. Qiu}, Complex Variables, Theory Appl. 30, No. 1, 77-96 (1996; Zbl 0854.30017)].