Sharp $\mathcal{SL}$- and $\mathcal{M}(\beta)$-radii for the class $\mathcal{F}_3$

Let $\mathcal{F}_3$ be the class of functions considered in the paper, and let $R_{\mathcal{SL}}$ and $R_{\mathcal{M}(\beta)}$ denote the radii for the corresponding classes, with parameter $\beta$. The sharp radius result for $\mathcal{F}_3$. The sharp $\mathcal{SL}$-radius and sharp $\mathcal{M}(\beta)$-radius for the class $\mathcal{F}_3$ are given by

$$R_{\mathcal{SL}}=\frac{3}{2}+\frac{3}{2\sqrt{2}}-\frac{1}{2}\sqrt{\frac{27}{2}+7\sqrt{2}}\simeq0.142009, \quad R_{\mathcal{M}(\beta)}=\frac{2(\beta-1)}{3+\sqrt{9+4(\beta-1)(\beta-2)}}.$$

The preceding arguments establish these constants by inclusion estimates for the logarithmic derivative and demonstrate sharpness using extremal functions from $\mathcal{F}_3$.