The Bounds of Some Determinants for Starlike Functions of Order Alpha

“…The proof of Theorem 2.1 was based on Lemma 1.3 which is useful for such computing. This lemma was also applied in [3] and [4] to find sharp bounds analogous as in Theorem 2.1 for the class of starlike functions of order α and of strongly starlike functions, respectively. Let now remark that the result of Theorem 2.1 can be achieved in a simple way by using Theorem 2.4 of [8] which particularly produces the following result: if p ∈ P is of the form (1.4) and µ ∈ [0, 1], then for n, m ∈ N, m < n, the following sharp estimate holds Theorem 2.4 can be found in [14] as Corollary 3.2.…”

Section: Sincementioning

confidence: 99%

The bounds of some determinants for functions of bounded turning of order alpha

Kowalczyk¹,

Kwon²,

Lecko³

et al. 2017

Bulletin

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“…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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The Bounds of Some Determinants for Starlike Functions of Order Alpha

Cited by 59 publications

References 9 publications

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

The bounds of some determinants for functions of bounded turning of order alpha

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

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