Sharp third Hankel determinant bounds for Ozaki close-to-convex function classes

Let $f\in\mathcal A$ be of the form

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

Here $H_3(1)$ denotes the third Hankel determinant formed from the coefficients of $f$, and $\mathcal F$ and $\mathcal G$ are the function classes defined in the paper.

Sharp-value conjecture. If $f\in\mathcal F$, then

$$|H_3(1)|\leq \frac{1}{16}=0.0625;$$

if $f\in\mathcal G$, then

$$|H_3(1)|\leq \frac{19}{2160}=0.00879\ldots.$$

Both estimates are sharp, with extremal functions $\frac{1+2z^2}{1-z^2}$ and $\frac{1}{2}\left(z\sqrt{1-z^2}+\arcsin z\right)$, respectively, obtained for $\omega(z)=z^2$ in the corresponding representations.

The conjecture proposes the sharp values improving the upper bounds established earlier in the paper. The stated extremal functions indicate equality cases for the two classes, but the supplied text does not establish the conjectured sharpness.