The sharp \$\\mathcal{S}^_{1/2}\$-radius of \$\\mathcal{S}^_e\$

Theorem. Let \$\\mathcal{S}^*_e=\\mathcal{S}^*(e^z)\$ and let \$\\mathcal{S}^*_{1/2}=\\mathcal{S}^*(1/(1-z))\$. The sharp directed radius is

$$ r^*=\\log 2\\approx 0.69314718055994530942. $$

For \$f\\in\\mathcal{S}^*_e\$, write \$p(z)=zf'(z)/f(z)=e^{\\omega(z)}\$, where \$\\omega(0)=0\$ and \$|\\omega(z)|\\le |z|\$ by Schwarz's lemma. The order-one-half inverse is \$\\varphi_{1/2}^{-1}(w)=(w-1)/w=1-1/w\$, so the target condition is controlled by

$$ \\psi(u)=\\varphi_{1/2}^{-1}(e^u)=1-e^{-u}. $$

For \$|u|=r\$, the exponential series and the triangle inequality give

$$ |\\psi(u)|=|e^{-u}-1| \\le \\sum_{n\\ge1}\\frac{|u|^n}{n!} =e^r-1=:B(r). $$

The majorant is strictly increasing, \$B'(r)=e^r>0\$, and \$B(\\log 2)=1\$. Thus \$\\psi(\\mathbb D_r)\\subseteq\\mathbb D\$ for \$r\\le\\log2\$. Equality holds at the negative real point \$u=-r\$, where \$\\psi(-r)=1-e^r\$, so the majorant is attained and the global containment bound is sharp.

Sharpness / extremal function. Take the Ma--Minda extremal $$ f_0(z)=z\\exp\\left(\\int_0^z\\frac{e^t-1}{t}\\,dt\\right), \\qquad \\frac{zf_0'(z)}{f_0(z)}=e^z. $$ For any \$r^*<R\\le1\$, its dilation \$f_{0,R}(z)=f_0(Rz)/R\$ remains in \$\\mathcal S^*_e\$. At \$z=-r^*/R\\in\\mathbb D\$, the logarithmic argument equals \$e^{-r^*}=1/2\$, and \$\\varphi_{1/2}^{-1}(1/2)=-1\$, a boundary point of the target disk. Hence no larger radius can hold.

Machine check. The executable verifier verify_global_max_axis_symbolic('exponential','order_0.5','log(2)') returns status: proven; it checks the composition, the exponential-series majorant at exact complex sample points, positivity of \$B'\$, and \$B(\\log2)=1\$. The non-algebraic premise is the elementary termwise triangle inequality.

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}^*_{1/2}\\)-radius of \\(\\mathcal{S}^*_e\\)

The sharp \\(\\mathcal{S}^_{1/2}\\)-radius of \\(\\mathcal{S}^_e\\)