The sharp \\(\\mathcal{S}^*_L\\)-radius of \\(\\mathcal{S}^*_e\\)

Theorem. For the exponential Ma--Minda class \\(\\mathcal{S}^*_e=\\mathcal{S}^*(e^z)\\), the sharp radius into the Bernoulli lemniscate class \\(\\mathcal{S}^*_L=\\mathcal{S}^*(\\sqrt{1+z})\\) is

$$ r^*=\\frac{\\log2}{2}\\approx0.34657359027997265471. $$

If \\(p(z)=e^{\\omega(z)}\\) with \\(|\\omega(z)|\\le|z|\\), put \\(u=\\omega(z)\\). The inverse \\(\\varphi_L^{-1}(w)=w^2-1\\) is branch-sensitive. For \\(|u|\\le r^*=\\log2/2<\\pi/2\\), \\(|\\operatorname{Im}u|<\\pi/2\\), hence \\(\\operatorname{Re}(e^u)>0\\). Thus \\(e^u\\) lies in the right lemniscate lobe and the principal square root satisfies \\(\\sqrt{e^{2u}}=e^u\\). Consequently \\(w^2-1\\) is the valid inverse on the full image considered here, and

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

The exponential series gives the global disk majorant

$$ |\\psi(u)|=|e^{2u}-1| \\le \\sum_{n\\ge1}\\frac{(2|u|)^n}{n!} =e^{2|u|}-1=:B(|u|). $$

Since \\(B'(r)=2e^{2r}>0\\) and \\(B(\\log2/2)=1\\), this proves \\(\\psi(\\mathbb D_r)\\subseteq\\mathbb D\\) for \\(r\\le r^*\\). Equality occurs at \\(u=r^*\\), where \\(e^{2r^*}=2\\) and \\(\\psi(u)=1\\), so the global bound is attained on the positive real axis.

Sharpness / extremal function. The extremal $$ f_0(z)=z\\exp\\left(\\int_0^z\\frac{e^t-1}{t}\\,dt\\right) $$ has \\(zf_0'(z)/f_0(z)=e^z\\). For every \\(r^*<R\\le1\\), the dilation \\(f_{0,R}(z)=f_0(Rz)/R\\) remains in \\(\\mathcal S^*_e\\). Take \\(z=r^*/R\\); then the argument is \\(e^{r^*}=\\sqrt2\\), and \\(\\varphi_L^{-1}(\\sqrt2)=1\\) lies on the target boundary. Therefore no larger radius is possible.

Machine check. verify_global_max_axis_symbolic('exponential','lemniscate','log(2)/2')returnsstatus: proven`, checking the candidate identity, the right-half-plane branch strip, the inverse composition, the exponential-series majorant at exact complex samples, the positive derivative, and the exact threshold. The analytic premises are the principal-square-root identity on \\(|\\operatorname{Im}u|<\\pi/2\\) and the elementary termwise triangle inequality.

Literature status. The local reconciliation row is touch_proven_exact / NO_EXTRACTED_CLAIM in the partial corpus. This is a machine-verified exact result, not a novelty claim; external literature audit and expert sign-off remain required.