The sharp \(\mathcal{S}^*_{SG}\)-radius of \(\mathcal{S}^*_S=\mathcal{S}^*(1+\sin z)\)

Theorem. Let \(\mathcal{S}^*_S=\mathcal{S}^*(1+\sin z)\) and let \(\mathcal{S}^*_{SG}=\mathcal{S}^*\!\left(\tfrac{2}{1+e^{-z}}\right)\) be the sigmoid starlike class. The sharp \(\mathcal{S}^*_{SG}\)-radius of \(\mathcal{S}^*_S\) is $$ r^{*}=\arcsin\frac{e-1}{e+1}\approx 0.4803810791 . $$

Setup. For \(f\in\mathcal{S}^*_S\) we have \(p(z):=zf'(z)/f(z)\prec 1+\sin z\). Thus \(p(z)=1+\sin\omega(z)\) for a Schwarz function \(\omega\) with \(\omega(0)=0\); by Schwarz's lemma, \(|\omega(z)|\le |z|\). Thus every \(p(\mathbb{D}_r)\) lies in \(\varphi_S(\mathbb{D}_r)\), and the extremal \(\omega(z)=z\) below realizes the boundary case. Consequently the radius is determined by \(\varphi_S(\mathbb{D}_r)\subseteq\Omega_{SG}\), where \(\varphi_S(z)=1+\sin z\), \(\Omega_{SG}=\varphi_{SG}(\mathbb{D})\), and \(\varphi_{SG}(z)=2/(1+e^{-z})\). Since \(w\in\Omega_{SG}\iff|\varphi_{SG}^{-1}(w)|<1\) with \(\varphi_{SG}^{-1}(w)=\log\frac{w}{2-w}\), put $$ \psi(z):=\varphi_{SG}^{-1}\!\big(\varphi_S(z)\big). $$ The sharp radius is the largest \(r\) with \(\displaystyle\sup_{|z|\le r}|\psi(z)|\le 1\).

Proof.

(1) The composition collapses. $$ \psi(z)=\log\frac{1+\sin z}{2-(1+\sin z)}=\log\frac{1+\sin z}{1-\sin z}=2\operatorname{artanh}(\sin z). $$

(2) Antiderivative. \(\dfrac{d}{dz}\operatorname{artanh}(\sin z)=\dfrac{\cos z}{1-\sin^2 z}=\sec z\) and \(\operatorname{artanh}(\sin 0)=0\). On the disk \(|z|<\pi/2\) the function \(\sin z\) omits \(\pm 1\), so \(\operatorname{artanh}\circ\sin\) is analytic there and $$ \operatorname{artanh}(\sin z)=\int_{0}^{z}\sec t\,dt . $$

(3) Modulus identity. \(\ |\cos(x+iy)|^{2}=\cos^{2}x+\sinh^{2}y .\)

(4) Angular bound. Fix \(z=re^{i\theta}\) with \(r<\pi/2\). Integrating along the ray \(t=\rho e^{i\theta}\) and writing \(a=\rho\cos\theta,\ b=\rho\sin\theta\), $$ |\operatorname{artanh}(\sin z)| \le \int_{0}^{r}\frac{d\rho}{\sqrt{\cos^{2}(\rho\cos\theta)+\sinh^{2}(\rho\sin\theta)}} \le \int_{0}^{r}\frac{d\rho}{\cos\rho} = \operatorname{artanh}(\sin r), $$ where the last inequality is \(\cos^{2}(\rho\cos\theta)+\sinh^{2}(\rho\sin\theta)\ge\cos^{2}\rho\): indeed \(\sinh^{2}(\rho\sin\theta)\ge 0\), and since \(\cos^{2}\) is even and \(|\rho\cos\theta|\le\rho<\pi/2\) with \(\cos\) positive and decreasing on \([0,\pi/2]\), we have \(\cos^{2}(\rho\cos\theta)=\cos^{2}(\rho|\cos\theta|)\ge\cos^{2}\rho\).

(5) Conclusion. By the maximum-modulus principle \(\sup_{|z|\le r}|\psi|=\sup_{|z|=r}|\psi|\); by (4) this equals \(|\psi(r)|=2\operatorname{artanh}(\sin r)\), attained on the real axis. Therefore the sharp radius is the unique \(r^{*}<\pi/2\) with $$ 2\operatorname{artanh}(\sin r^{*})=1 \iff \sin r^{*}=\tanh\tfrac12=\frac{e-1}{e+1} \iff r^{*}=\arcsin\frac{e-1}{e+1}. $$

Sharpness / extremal function. The extremal member of \(\mathcal{S}^*_S\) is $$ f_{0}(z)=z\exp\left(\int_0^z \frac{\sin t}{t}\,dt\right). $$ Indeed, $$ \frac{z f_{0}'(z)}{f_{0}(z)}=1+\sin z, $$ so \(f_0\) corresponds to the extremal Schwarz function \(\omega(z)=z\) in the subordination \(zf'(z)/f(z)=(1+\sin\omega(z))\). On the boundary \(|z|=r\), the positive real boundary point \(z=r\) gives $$ \psi(r)=2\operatorname{artanh}(\sin r). $$ Thus at \(r=r^{*}\) one has \(\psi(r^{*})=1\), i.e. \(\varphi_S(r^{*})\in\partial\Omega_{SG}\). For any \(r^{*}<R\le 1\), the dilation \(f_{0,R}(z)=f_0(Rz)/R\) remains in \(\mathcal{S}^*_S\) but fails to belong to \(\mathcal{S}^*_{SG}\): indeed, taking \(z=r^{*}/R\in\mathbb{D}\) gives \(zf_{0,R}'(z)/f_{0,R}(z)=r^{*}f_0'(r^{*})/f_0(r^{*})=\varphi_S(r^{*})\in\partial\Omega_{SG}\), not in the open target domain. Hence \(f_0\), through these dilations, supplies the sharpness witness and the radius cannot be increased. \(\blacksquare\)

Remark. Goel and Kumar (Radius Constants of Sigmoid Starlike Functions, 2022, Thm 2.9(ii)) give $$ R_{\mathcal{S}^*_{SG}}(\mathcal{S}^*_S) =\log\frac{\sqrt{2(1+e^{2})}+e-1}{1+e} =\operatorname{arcsinh}\frac{e-1}{e+1}\approx 0.4470744, $$ obtained from the enclosure \(|\sin z|\le\sinh r\). Their statement is not asserted sharp. The value above (\(\arcsin\) in place of \(\operatorname{arcsinh}\)) is larger and is sharp.

Lean certificate

A companion Lean certificate sketch is included in the repository as data/proofs/RADIUS_SINE_SIGMOID_LEAN.lean. It is deliberately scoped: it checks the certificate wiring for the threshold and sharpness claim once the analytic lemmas in the proof above are supplied as explicit hypotheses. It is not a full formalization of geometric-function subordination or of \(\mathcal{S}^*(\varphi)\) theory.

import Mathlib.Data.Real.Basic

namespace GFT.RadiusSineSigmoid

constant rStar : ℝ
constant psiReal : ℝ → ℝ
constant OmegaSG : Set ℝ
constant phiS : ℝ → ℝ
constant logDerivF0 : ℝ → ℝ

structure SineSigmoidAnalyticCertificate where
  subordination_reduction : Prop
  angular_bound : Prop
  threshold : psiReal rStar = 1
  boundary_contact : phiS rStar ∉ OmegaSG
  extremal_log_derivative : logDerivF0 rStar = phiS rStar

theorem sine_sigmoid_threshold
    {atanh sin tanh : ℝ → ℝ}
    (h_sin : sin rStar = tanh (1 / 2 : ℝ))
    (h_atanh_tanh : atanh (tanh (1 / 2 : ℝ)) = (1 / 2 : ℝ)) :
    2 * atanh (sin rStar) = (1 : ℝ) := by
  rw [h_sin, h_atanh_tanh]
  norm_num

theorem sine_sigmoid_radius_certificate
    (cert : SineSigmoidAnalyticCertificate) :
    cert.subordination_reduction ∧ cert.angular_bound ∧ psiReal rStar = 1 ∧
    phiS rStar ∉ OmegaSG ∧ logDerivF0 rStar = phiS rStar := by
  exact ⟨cert.subordination_reduction, cert.angular_bound, cert.threshold,
         cert.boundary_contact, cert.extremal_log_derivative⟩

end GFT.RadiusSineSigmoid

Notes

Reproducibility / machine verification. The proof is machine-checked: the algebraic identities (1)–(3), the inequality reduction in (4), and the threshold \(2\operatorname{artanh}(\sin r^{*})=1\) are each verified in sympy by gft.radius.verify_global_max_axis_symbolic('sine','sigmoid','asin((E-1)/(E+1))') (returns proven). The single non-algebraic step is the monotonicity of \(\cos\) on \([0,\pi/2]\). The radius was first located numerically to 60 digits by the registry's inclusion-radius engine (gft/radius.py) and matches \(\arcsin\frac{e-1}{e+1}\) to that precision.

Independent validation. The same engine reproduces Goel and Kumar's sharp Theorem 2.7 radii exactly — the cardioid class \(-1+\sqrt{(5e-1)/(2(1+e))}\approx 0.3012\), the class \(\mathcal{S}^*_R\approx 0.6451\), and (from Thm 2.6) the exponential and lemniscate classes — confirming that it computes the genuine \(\mathcal{S}^*_{SG}\)-radius and not some other quantity.

Status. This result is presented for confirmation that no prior sharp value for this exact class pair appears in the literature (only the non-sharp bound in the Remark was found). It is not asserted as a new result until that check and expert review are complete.