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.

For analytic functionsfin the unit disk $${\mathbb {D}}$$D normalized by $$f(0)=0$$f(0)=0 and $$f'(0)=1$$f′(0)=1 satisfying in $${\mathbb {D}}$$D respectively the conditions $${{\,\mathrm{Re}\,}}\{ (1-z)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z+z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z)^2f'(z) \} > 0,$$Re{(1-z)f′(z)}>0,Re{(1-z2)f′(z)}>0,Re{(1-z+z2)f′(z)}>0,Re{(1-z)2f′(z)}>0, the sharp upper bound of the third logarithmic coefficient in case when $$f''(0)$$f′′(0) is real was computed.

Let f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$D = { z ∈ C : | z | < 1 } , and $${{\mathcal {S}}}$$S be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$z ∈ D . We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$H 2 ( 2 ) ( f ) = a 2 a 4 - a 3 2 for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$F O ( λ , β ) of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$1 / 2 ≤ λ ≤ 1 , and $$0<\beta \le 1$$0 < β ≤ 1 . Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$| H 2 ( 2 ) ( f - 1 ) | , where $$f^{-1}$$f - 1 is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$| H 2 ( 2 ) ( f ) | and $$|H_2(2)(f^{-1})|$$| H 2 ( 2 ) ( f - 1 ) | for strongly convex functions.