Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha

“…We will find also the sharp bound of the second Hankel determinant H 2,2 (f ) = a 2 a 4 − a 2 3 . Both functionals J 2,3 and H 2,2 have been studied recently by various authors (see e.g., [5,6,14,16,17,19,27]).…”

Section: Zalcman Functional and Hankel Determinantmentioning

confidence: 99%

Coefficient Problems in the Subclasses of Close-to-Star Functions

Lecko

Sim

2019

Results Math

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For two subclasses of close-to-star functions we estimate early logarithmic coefficients, coefficients of inverse functions, Hankel determinant H2,2 and Zalcman functional J2,3. All results are sharp. Mathematics Subject Classification. 30C45. Keywords. Univalent functions, close-to-star functions, functions starlike in the direction of the real axis, functions convex in the direction of the imaginary axis, Hankel determinant, Zalcman functional, logarithmic coefficients, inverse functions, Carathéodory class. Denote by CST the class of all close-to-star functions introduced by Reade [30]. Note that f ∈ CST if and only if a function F (z) := z 0 f (t) t dt, z ∈ D, (1.3) is close-to-convex [15], [12, Vol. II, p. 3]. The class of close-to-star functions and its subclasses were intensively studied by various authors (e.g., MacGregor [25], Sakaguchi [32], Causey and Merkes [4]; for further references, see [12, Vol. II, pp. 97-104]). Given g ∈ S * and β ∈ R, let CST β (g) be the subclass of CST of all f satisfying (1.2). The classes CST 0are particularly interesting and were separately studied by authors. In this paper we deal with the classes CST 0 (g 1 ) =: ST (i) and CST 0 (g 2 ) =: ST (1) which elements f in view of (1.2) satisfy the conditionandRe5) respectively. Let us add the inequality (1.4) defines the subclass of the class of functions starlike in the direction of the real axis introduced by Robertson [31]. Moreover, each function F given by (1.3) over the class ST (i) maps univalently D onto a domain F (D) convex in the direction of the imaginary axis. The concept of convexity in one direction belongs to Roberston [31] (see e.g., [12, p. 199]). Each function F given by (1.3) over the class ST (1) maps univalently D onto a domain F (D) called convex in the positive the direction of the real axis, i.e., {w + it: t ≥ 0} ⊂ f (D) for every w ∈ f (D) [2,8,9,11,20,21]. Let us remark that the condition (1.4) was generalized by replacing the expression 1 − z 2 by the expression 1 − α 2 z 2 with α ∈ [0, 1] in [13].In this paper we find the sharp estimates of early logarithmic coefficients (Sect. 2), of the Hankel determinant H 2,2 and of Zalcman functional J 2,3 (Sect. 3) and of the early inverse coefficients (Sect. 4) of functions in the classes ST (i) and ST (1). Since both classes ST (i) and ST (1) have a representation using the Carathéodory class P, i.e., the class of functions p ∈ H of the form(1.6) having a positive real part in D, the coefficients of functions in ST (i) and ST (1) have a suitable representation expressed by the coefficients of functions in P. Therefore to get the upper bounds of considered functionals our computing is based on parametric formulas for the second and third coefficients Vol. 74 (2019) Coefficients Problems

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“…In recent years, the research on Hankel determinants has focused on the estimation of |H 2 (2)|. Problems in this field has also been argued by several authors for various classes of univalent functions [14][15][16][17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

The Second Hankel Determinant Problem for a Class of Bi-Close-to-Convex Functions

et al. 2019

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Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha

Cited by 68 publications

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The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions of Order 1 / 2

The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions of Order 1 / 2

Coefficient Problems in the Subclasses of Close-to-Star Functions

The Second Hankel Determinant Problem for a Class of Bi-Close-to-Convex Functions

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