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

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