The second Hankel determinant for strongly convex and Ozaki close-to-convex functions

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