Third Hankel determinant for starlike and convex functions associated with the exponential function

Lemma 1. Let \(p\in {\mathcal {P}}\), and be given by ([3\)](file:///article/10.1007/s40590-024-00695-4#Equ3), then \$\$\begin{aligned} 2c\_{2}= & c\_{1}^{2}+\delta (4-c\_{1}^{2}), \end{aligned}\$\$ (4) \$\$\begin{aligned} 4c\_{3}= & c\_{1}^{3}+2(4-c\_{1}^{2})c\_{1}\delta -(4-c\_{1} ^{2})c\_{1}\delta ^{2}+2(4-c\_{1}^{2})(1-\left| \delta \right| ^2)\eta , \end{aligned}\$\$ (5) \$\$\begin{aligned} 8c\_{4}= & c\_{1}^{4}+(4-c\_{1}^{2})\delta (c\_{1}^{2}(\delta ^{2}-3\delta +3)+4\delta )-4(4-c\_{1}^{2})(1-\left| \delta \right| ^{2})(c\_{1}(\delta -1)\eta \nonumber \ \ & +{\bar{\delta }}\eta ^{2}-(1-\left| \eta \right| ^{2})\rho ), \end{aligned}\$\$ (6) for some \(\rho \), \(\delta \) and \(\eta \) such that \(\left| \rho \right| \le 1\), \(\left| \delta \right| \le 1\) and \(\left| \eta \right| \le 1\).

Theorem 2. Let \(f\in {\mathcal {S}}^{\*}\_{e}\) and be given by ([1](file:///article/10.1007/s40590-024-00695-4#Equ1)). Then \$\$\begin{aligned} \left| H\_{3,1}(f)\right| \le \frac{1}{9}. \end{aligned}\$\$ (7) The inequality is sharp for the function \(f\_{0}\) defined by \$\$\begin{aligned} f\_{0}(z)=z\exp \left( \int \_{0}^{z}\frac{e^{t^{3}}-1}{t}\textrm{d} t\right) =z+\frac{1}{ 3}z^{4}+\cdots . \end{aligned}\$\$ (8)

Theorem 3. Let \(f\in {\mathcal {C}}\_{e}\) and be given by ([1\)](file:///article/10.1007/s40590-024-00695-4#Equ1). Then \$\$\begin{aligned} \left| H\_{3,1}(f)\right| \le \frac{1}{144}. \end{aligned}\$\$ (16) The inequality is sharp for \(f\_{1}\) given by \$\$\begin{aligned} f\_{1}(z)=\int \_{0}^{z}\left( \exp \left( \int \_{0}^{x}\frac{e^{t^{3}}-1}{t} \textrm{d}t\right) \right) \textrm{d}x=z+\frac{1}{12}z^{4}+\cdots . \end{aligned}\$\$ (17)