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In the present paper, we investigated some conditions to be in the class of Carathéodory functions by using the concept of the first-order differential subordinations. Moreover, various interesting special cases were considered in the geometric function theory as applications of main results presented here. [ABSTRACT FROM AUTHOR]

In recent years, the study of Hankel determinants for various subclasses of normalised univalent functions f ∈ S given by f (z) = z + ∑ n = 2 ∞ a n z n for D = { z ∈ 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 (f) = a 2 a 4 − a 3 2 . A non-sharp bound for H 2 , 2 (f) when f ∈ K (α) , α ∈ [ 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 (f − 1) for the inverse functions f − 1 when f ∈ K (α) , and when f ∈ S ∗ (α) , 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 S ∗ (α) , K (α) , S β ∗ and K β , where S β ∗ and K β are, respectively, the classes of strongly starlike, and strongly convex functions of order β. [ABSTRACT FROM AUTHOR]

Let $f$ be analytic in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z| and ${\mathcal{S}}$ be the subclass of normalised univalent functions given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ for $z\in \mathbb{D}$. We give sharp upper and lower bounds for $|a_{3}|-|a_{2}|$ and other related functionals for the subclass ${\mathcal{F}}_{O}(\unicode[STIX]{x1D706})$ of Ozaki close-to-convex functions. [ABSTRACT FROM AUTHOR]

In the present paper, the estimate of the third Hankel determinant H 3 , 1 (f) = a 1 a 2 a 3 a 2 a 3 a 4 a 3 a 4 a 5 for the class of starlike functions, i.e., for the class of analytic functions f standardly normalized such that Re (z f ′ (z) / f (z)) > 0 , z ∈ D : = { z ∈ C : | z | < 1 } , is improved. [ABSTRACT FROM AUTHOR]