On the difference of the two initial logarithmic coefficients for Bazilevic class of univalent functions

Theorem 1. Let $f \in \mathcal{B}_1(\alpha)$ for some $\alpha > 0$ . Then $$-\frac{1}{\sqrt{(\alpha+1)^2+1}} \le |\gamma_2| - |\gamma_1| \le \frac{1}{\alpha+2}.$$ The right estimation is sharp for all $\alpha > 0$ , while the left is sharp for $\alpha_1 \le \alpha \le \alpha_2$ , where $\alpha_1 = \frac{1}{2} \left( \sqrt{6} - 2 \right) = 0.2247 \dots$ and $\alpha_2 = \sqrt{2} - 1 = 0.4142 \dots$ are the positive roots of the equation $$2\alpha^4 + 8\alpha^3 + 5\alpha^2 - 6\alpha + 1 = 0.$$