We prove the sharp inequality $|H_{3,1}(f)|\leq 4/135$ for convex functions, that is, for analytic functions $f$ with $a_{n}:=f^{(n)}(0)/n!,~n\in \mathbb{N}$ , such that $$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\end{eqnarray}$$ where $H_{3,1}(f)$ is the third Hankel determinant $$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$

In the present paper, we proved the sharp inequality \(|H_{3,1}(f)|\le 1/9\) for analytic functions f with \(a_n:=f^{(n)}(0)/n!,\ n\in {\mathbb {N}},\ a_1:=1,\) such that $$\begin{aligned} {{\mathrm{Re}}}\frac{zf'(z)}{f(z)}> \frac{1}{2},\quad z\in {\mathbb {D}}:=\{z \in {\mathbb {C}} : |z|<1\}, \end{aligned}$$ where $$\begin{aligned} H_{3,1}(f):= \begin{vmatrix} a_1&\quad a_2&\quad a_3 \\ a_2&\quad a_3&\quad a_4 \\ a_3&\quad a_4&\quad a_5 \end{vmatrix} \end{aligned}$$ is the third Hankel determinant.

In the present paper, the estimate of the third Hankel determinant $$\begin{aligned} \begin{aligned} H_{3,1}(f)&= \begin{vmatrix} a_{1}&a_{2}&a_{3} \\ a_{2}&a_{3}&a_{4} \\ a_{3}&a_{4}&a_{5} \end{vmatrix} \end{aligned} \end{aligned}$$ for the class of starlike functions, i.e., for the class of analytic functions f standardly normalized such that \({{\mathrm{Re}}}(zf'(z)/f(z)) > 0,\ z\in {{\mathbb {D}}}:=\{z \in {\mathbb {C}} : |z|<1\},\) is improved.

We begin the study of Hankel matrices whose entries are logarithmic coefficients of univalent functions and give sharp bounds for the second Hankel determinant of logarithmic coefficients of convex and starlike functions.