Assume that $\gamma$ is a starlike curve in the complex plane w.r.t. 0. Let $\alpha\in (0,\pi/2]$. For each $z\in\gamma$, $z\neq x$, we let $\alpha(z,x)$ denote the acute angle which the segment $[z,x]$ makes with the ray from $0$ through $x$, and we define $$\alpha(x)=\liminf_{z\rightarrow x}\alpha(z, x)\in [0,\pi/2].$$ Assume that $\alpha(x)\ge \alpha$. My question arises, is this true $$\alpha_1(x)=\liminf_{z,w\rightarrow x}\alpha(z, w)>0.$$

Assume that $\gamma$ is a starlike Jordan curve in the complex plane w.r.t. 0. Let $\alpha\in (0,\pi/2]$. For each $z\in\gamma$, $z\neq x$, we let $\alpha(z,x)$ denote the acute angle which the segment $[z,x]$ makes with the ray from $0$ through $x$, and we define $$\alpha(x)=\liminf_{z\rightarrow x}\alpha(z, x)\in [0,\pi/2].$$ Assume that $\alpha(x)\ge \alpha$. My question arises, is this true $$\alpha_1(x)=\liminf_{z,w\rightarrow x}\alpha(z, w)>0.$$