The sharp \$\\mathcal{S}^_{3/4}\$-radius of \$\\mathcal S^\$

Theorem. Let \$\\mathcal S^*=\\mathcal S^*((1+z)/(1-z))\$ and \$\\mathcal S^*_{3/4}=\\mathcal S^*((1-z/2)/(1-z))\$, the order-3/4 Ma--Minda target. Its inverse is \$\\varphi_{3/4}^{-1}(w)=(w-1)/(w-1/2)\$. The sharp radius is

$$ r^*=\\frac17\\approx0.14285714285714285714. $$

For \$p(z)=\\varphi_S(\\omega(z))\$, Schwarz's lemma reduces the radius problem to \$u=\\omega(z)\$ and \$|u|\\le|z|\$. The target inverse is \$\\varphi_{3/4}^{-1}(w)=(w-1)/(w-1/2)\$, hence

$$ \\psi(u)=\\frac{\\varphi_S(u)-1}{\\varphi_S(u)-1/2} =\\frac{4u}{1+3u}. $$

For \$|u|=r<1/3\$, the reverse triangle inequality yields

$$ |\\psi(u)|=\\frac{4r}{|1+3u|} \\le\\frac{4r}{1-3r}=:G(r). $$

The majorant is strictly increasing, since \$G'(r)=4/(1-3r)^2>0\$. At \$r^*=1/7\$, \$G(r^*)=1\$. Thus the entire disk of radius \$1/7\$ is contained in the target preimage, and equality is attained at \$u=-1/7\$, where the reverse triangle inequality is exact.

Sharpness / extremal function. The Koebe extremal $$ f_0(z)=\\frac{z}{(1-z)^2},\\qquad \\frac{zf_0'(z)}{f_0(z)}=\\frac{1+z}{1-z} $$ realizes \$\\omega(z)=z\$. For every \$1/7<R\\le1\$, the dilation \$f_{0,R}(z)=f_0(Rz)/R\$ remains starlike. At \$z=-1/(7R)\$, its logarithmic argument is \$\\varphi_S(-1/7)=3/4\$, and the target inverse is \$-1\$, a boundary point. Therefore no larger radius is possible.

Machine check. verify_global_max_axis_symbolic('starlike','order_0.75','1/7') returns status: proven; it checks the rational composition and discharges the reverse-triangle guard symbolically through \$|1+3re^{it}|^2-(1-3r)^2=6r(1+\\cos t)=12r\\cos^2(t/2)\\ge0\$, then checks positivity of \$G'\$ and \$G(1/7)=1\$. Thus this executable guard is an exact identity rather than a finite sample check.

Literature status. The local reconciliation snapshot labels this directed row KNOWN_GENERAL as a standard order-\$\\alpha\$ inclusion result, with citation confirmation pending. This is a machine-verified exact result, not a novelty claim; external literature audit and expert sign-off remain required.

The sharp \\(\\mathcal{S}^*_{3/4}\\)-radius of \\(\\mathcal S^*\\)

The sharp \\(\\mathcal{S}^_{3/4}\\)-radius of \\(\\mathcal S^\\)