Kumar–Gangania conjecture on the third Hankel determinant for cardioid starlike functions

Let $\mathcal{S}_{\wp}^{*}$ be the class of functions $f$ analytic in the unit disk, normalized by $f(0)=0$ and $f'(0)=1$, such that

$$\frac{zf'(z)}{f(z)}\prec 1+ze^z.$$

For $f(z)=z+a_2z^2+a_3z^3+\cdots$, let $H_3(1)$ denote the third Hankel determinant

$$H_3(1)=\begin{vmatrix}a_1&a_2&a_3\\a_2&a_3&a_4\\a_3&a_4&a_5\end{vmatrix},\qquad a_1=1.$$

Kumar–Gangania conjecture. If $f\in\mathcal{S}_{\wp}^{*}$, then the sharp bound is

$$|H_3(1)|\leq \frac{1}{9}\approx 0.1111\ldots,$$

with extremal function

$$f(z)=z\exp\left(\frac{1}{3}(e^{z^3}-1)\right)=z+\frac{1}{3}z^4+\frac{2}{9}z^7+\cdots.$$

Kumar and Gangania had previously obtained the weaker bound $|H_3(1)|\leq 0.150627$ for this class; the conjecture proposes the sharp value and an extremal function. Its resolution is not supplied in the source material.