The radial stretch is quasiconformal with K equal to max of alpha and one over alpha

Given: Choice, $\alpha >0$, the ACL/Sobolev convention, and the annulus path-family conventions.

The map has polar form $f_{\alpha }(r{e}^{it})=r^{\alpha }{e}^{it}$. The function $r\mapsto r^{\alpha }$ is a strictly increasing homeomorphism of $[0,\mathrm{\infty })$ with inverse $s\mapsto s^{1\mathrm{/}\alpha }$, so $f_{\alpha }$ is a homeomorphism with inverse $f_{1\mathrm{/}\alpha }$.

On $\mathbb{C}\setminus \{0\}$, direct Wirtinger differentiation gives the derivatives in the Statement. In particular the real Jacobian is $J_{f_{\alpha }}=\mathrm{\mid }(f_{\alpha }{)}_z{\mathrm{\mid }}^2-\mathrm{\mid }(f_{\alpha }{)}_{\bar{z}}{\mathrm{\mid }}^2=\alpha \mathrm{\mid }z{\mathrm{\mid }}^{2\alpha -2}>0.$

On each horizontal or vertical line not passing through $0$, $f_{\alpha }$ is smooth. On the two coordinate lines through $0$, its components are constant multiples of $g(t)=\mathrm{sgn}(t)\mathrm{\mid }t{\mathrm{\mid }}^{\alpha }$, which is absolutely continuous on compact intervals because $g^{\mathrm{\prime }}(t)=\alpha \mathrm{\mid }t{\mathrm{\mid }}^{\alpha -1}\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ for $\alpha >0$.

The function is locally bounded, and its classical first partial derivatives off $0$ are bounded by $C_{\alpha }\mathrm{\mid }z{\mathrm{\mid }}^{\alpha -1}$. Since ${\int }_{\mathrm{\mid }z\mathrm{\mid }<R}\mathrm{\mid }z{\mathrm{\mid }}^{2\alpha -2}dA=2\pi {\int }_0^R{r}^{2\alpha -1}dr<\mathrm{\infty }$ for $\alpha >0$, they are locally square-integrable (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma). The excluded point $0$ is null because it lies in boxes of arbitrarily small area (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included). The ACL characterization therefore gives $f_{\alpha }\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}$ and identifies these almost-everywhere classical derivatives with its weak derivatives (Absolute continuity on almost every coordinate line, The ACL characterisation of $W^{1,p}$).

The ratio $\mathrm{\mid }(f_{\alpha }{)}_{\bar{z}}\mathrm{\mid }\mathrm{/}\mathrm{\mid }(f_{\alpha }{)}_z\mathrm{\mid }=\mathrm{\mid }\alpha -1\mathrm{\mid }\mathrm{/}(\alpha +1)<1$ off $0$, and $\frac{1+\mathrm{\mid }\alpha -1\mathrm{\mid }\mathrm{/}(\alpha +1)}{1-\mathrm{\mid }\alpha -1\mathrm{\mid }\mathrm{/}(\alpha +1)}=\max (\alpha ,\frac{1}{\alpha })\mathrm{.}$ The modulus-distortion lemma gives the quadrilateral inequalities for analytic maps; together with orientation preservation this is the geometric definition (An analytically quasiconformal homeomorphism distorts quadrilateral moduli by at most K, Orientation-preserving homeomorphisms and the geometric definition of quasiconformality).

For every finite round annulus, $\lambda ({\mathrm{\Gamma }}_{r,R})=(2\pi {)}^{-1}\log (R\mathrm{/}r)$ and $\mu ({\mathrm{\Gamma }}_{r,R})=1\mathrm{/}\lambda ({\mathrm{\Gamma }}_{r,R})$ (Extremal length of the rectangle and of the round annulus).

At a point where the real derivative is invertible, the inverse-function theorem makes the map a local diffeomorphism; for a smooth local diffeomorphism its local-homology orientation multiplier is the sign of its determinant (The Euclidean inverse function theorem, Smooth orientation sign is the local integral homology multiplier, R-orientation of a topological manifold).