Let ${\mathcal{S}}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ which are of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ . We determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in ${\mathcal{S}}$ and certain of its subclasses. We also discuss similar problems for typically real functions.

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.

Let f be analytic in $$\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}$$D = { z ∈ C : | z | < 1 } , and be given by $$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$$f ( z ) = z + ∑ n = 2 ∞ a n z n . We give sharp bounds for the second Hankel determinant, some Toeplitz, and some Hermitian-Toeplitz determinants of functions in the class of Ozaki close-to-convex functions, together with a sharp bound for the Zalcman functional $$J_{2,3}(f).$$J 2 , 3 ( f ) .

In recent years, the study of Hankel determinants for various subclasses of normalised univalent functions $f\in \mathcal{S}$ given by $f\left(z\right)=z+\sum _{n=2}^{\mathrm{\infty }}{a}_n{z}^n$ for $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ has produced many interesting results. The main focus of interest has been estimating the second Hankel determinant of the form $H_{2,2}\left(f\right)=a_2{a}_4-a_3^2$. A non-sharp bound for $H_{2,2}\left(f\right)$ when $f\in \mathcal{K}\left(\alpha \right)$, $\alpha \in [0,1)$ consisting of convex functions of order α was found by Krishna and Ramreddy (Hankel determinant for starlike and convex functions of order alpha. Tbil Math J. 2012;5:65–76), and later improved by Thomas et al. (Univalent functions: a primer. Berlin: De Gruyter; 2018). In this paper, we give the sharp result. Moreover, we obtain sharp results for $H_{2,2}\left(f^{-1}\right)$ for the inverse functions $f^{-1}$ when $f\in \mathcal{K}\left(\alpha \right)$, and when $f\in {\mathcal{S}}^{\ast }\left(\alpha \right)$, the class of starlike functions of order α. Thus, the results in this paper complete the set of problems for the second Hankel determinants of f and $f^{-1}$ for the classes ${\mathcal{S}}^{\ast }\left(\alpha \right)$, $\mathcal{K}\left(\alpha \right)$, ${\mathcal{S}}_{\text{β}}^{\ast }$ and ${\mathcal{K}}_{\text{β}}$, where ${\mathcal{S}}_{\text{β}}^{\ast }$ and ${\mathcal{K}}_{\text{β}}$ are, respectively, the classes of strongly starlike, and strongly convex functions of order β.

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 with $f(0)=0$, and $f^{\prime }(0)=1.$ Let F be the inverse function of f, given by $F(z)=\omega +{\sum }_{n=2}^{\infty }{A}_n{\omega }^n$ for some $|\omega |\le r_0(f)$. Let ${\mathcal{S}}^{*}\subset \mathcal{S}$ be the subset of starlike functions in $\mathbb{D}$, and $\mathcal{C}$ the subset of convex functions in $\mathbb{D}.$ We show that $-1\le |A_3|-|A_2|\le 3$ for $f\in \mathcal{S}$, the upper bound being sharp, and sharp upper and lower bounds for $|A_3|-|A_2|$ for the more important subclasses of ${\mathcal{S}}^{*}$ and $\mathcal{C}$, and for some related classes of Bazilevič functions.

Let the function be analytic in and given by . For , denote by the class of strongly convex functions. We give sharp bounds for the initial coefficients of the inverse function of , showing that these estimates are the same as those for functions in , thus extending a classical result for convex functions. We also give invariance results for the second Hankel determinant , the first three coefficients of and Fekete–Szegö theorems.