Fekete–Szegő conjecture for the class of q-starlike functions of order alpha

Let $q$ satisfy $0<q<1$, let $0\leq\alpha<1$, and let $f\in\mathcal{S}_q^*(\alpha)$ have the normalized expansion

$$f(z)=z+a_2z^2+a_3z^3+\cdots.$$

Let $\mu$ be any complex number. Fekete–Szegő conjecture.

$$|a_3-\mu a_2^2|\leq \max\left\{\left|2(1-2\mu)\left(\frac{\ln \frac{q}{1-\alpha(1-q)}}{q-1}\right)^2+2\left(\frac{\ln \frac{q}{1-\alpha(1-q)}}{q^2-1}\right)\right|,2\left(\frac{\ln \frac{q}{1-\alpha(1-q)}}{q^2-1}\right)\right\}.$$

Equality occurs for the functions $F_1$ and $F_2$ given by

$$F_1(z)=z\exp\left[\sum_{n=1}^\infty \frac{2\ln \frac{q}{1-\alpha(1-q)}}{q^n-1}z^n\right]$$

and

$$F_2(z)=z\exp\left[\sum_{n=1}^\infty \frac{2\ln \frac{q}{1-\alpha(1-q)}}{q^{2n}-1}z^{2n}\right].$$

This is a Fekete–Szegő coefficient problem for the class $\mathcal{S}_q^*(\alpha)$; the stated extremal functions are intended to show sharpness, but the source presents the result as a conjecture and gives no resolution evidence.