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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]

In this paper, the authors derive several sufficient conditions for a function to be the Carathéodory function in the unit disk 𝔻: = {z ∈ ℂ: |z| < 1}. More precisely, for given β ∈ (–π/2, π/2), γ ∈ [0, cosβ) and δ ∈ (0, π/2], we find some sufficient conditions for an analytic function p such that p(0) = 1 to satisfy Re{e−iβp(z)} > γ or | arg {p(z)–γ} | < δ for all z ∈ 𝔻 by using the first-order differential subordination. We then apply the results obtained here in order to find some conditions for univalent functions with geometric properties such as spirallikeness and strongly starlikeness. [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]

Let f be analytic in the unit disk D = { z ∈ C : | z | < 1 } , and S be the subclass of normalized univalent functions with f (0) = 0 , and f ′ (0) = 1. Let F be the inverse function of f, given by F (z) = ω + ∑ n = 2 ∞ A n ω n for some | ω | ≤ r 0 (f) . Let S * ⊂ S be the subset of starlike functions in D , and C the subset of convex functions in D. We show that − 1 ≤ | A 3 | − | A 2 | ≤ 3 for f ∈ S , the upper bound being sharp, and sharp upper and lower bounds for | A 3 | − | A 2 | for the more important subclasses of S * and C , and for some related classes of Bazilevič functions. [ABSTRACT FROM AUTHOR]

Let Σ be the class of meromorphic functions f of the form f (ζ) = ζ + ∑ n = 0 ∞ a n ζ − n which are analytic in Δ : = { ζ ∈ C : | ζ | > 1 } . For n ∈ N 0 : = N ∪ { 0 } , the nth Faber polynomial Φ n (w) of f ∈ Σ is a monic polynomial of degree n that is generated by a function ζ f ′ (ζ) / (f (ζ) − w) . For given f ∈ Σ , by F n , i (f) , we denote the ith coefficient of Φ n (w) . For given 0 ≤ α < 1 and 0 < β ≤ 1 , let us consider domains H α and S β ⊂ C defined by H α = { w ∈ C : Re (w) > α } and S β = { w ∈ C : | arg (w) | < β } , which are symmetric with respect to the real axis. A function f ∈ Σ is called meromorphic starlike of order α if ζ f ′ (ζ) / f (ζ) ∈ H α for all ζ ∈ Δ . Another function f ∈ Σ is called meromorphic strongly starlike of order β if ζ f ′ (ζ) / f (ζ) ∈ S β for all ζ ∈ Δ . In this paper we investigate the sharp bounds of F n , n − i (f) , n ∈ N 0 , i ∈ { 2 , 3 , 4 } , for meromorphic starlike functions of order α and meromorphic strongly starlike of order β. Similar estimates for meromorphic convex functions of order α ( 0 ≤ α < 1 ) and meromorphic strongly convex of order β ( 0 < β ≤ 1 ) are also discussed. [ABSTRACT FROM AUTHOR]

Let SR * be the class of starlike functions with real coefficients, i.e., the class of analytic functions f which satisfy the condition f (0) = 0 = f ′ (0) − 1 , Re { z f ′ (z) / f (z) } > 0 , for z ∈ D : = { z ∈ C : | z | < 1 } and a n : = f (n) (0) / n ! is real for all n ∈ N . In the present paper, it is obtained that the sharp inequalities − 4 / 9 ≤ H 3 , 1 (f) ≤ 3 / 9 hold for f ∈ SR * , where H 3 , 1 (f) is the third Hankel determinant of order 3 defined by H 3 , 1 (f) = a 3 (a 2 a 4 − a 3 2) − a 4 (a 4 − a 2 a 3) + a 5 (a 3 − a 2 2) . [ABSTRACT FROM AUTHOR]

In the present paper, we find sufficient conditions for starlikeness and convexity of normalized Lommel functions of the first kind using the admissible function methods. Additionally, we investigate some inclusion relationships for various classes associated with the Lommel functions. The functions belonging to these classes are related to the starlike functions, convex functions, close-to-convex functions and quasi-convex functions. [ABSTRACT FROM AUTHOR]