Global convergence and monotonicity of Newton iteration for the inverse Gr\"otzsch modulus

Abstract

Let \[ \mu(r)=\frac{\pi}{2}\frac{\Kc(r')}{\Kc(r)}, \qquad r'=\sqrt{1-r^2},\qquad 0<r<1, \] be the Gr\"otzsch modulus function. Problems 3.38(a)--(b) in a survey of Vuorinen ask whether Newton's iteration for $\mu^{-1}(y)$, initialized by $x_0=1/\cosh y$, converges for every $y>\pi/2$, and whether it is strictly increasing when $y>\pi$. We prove both assertions. The main point is that $\mu$ has exactly one inflection point $a\in(1/2,1/\sqrt2)$. If the zero lies in the convex region, the Newton iterates increase to it from the left. If the zero lies in the concave region, the orbit crosses the inflection point, overshoots the zero at most once, and then decreases to the zero. For $y>\pi$ we obtain the stronger estimate $0<x_n<x_{n+1}<\mu^{-1}(y)<3-2\sqrt2<1.$