Sharp Bounds of Some Coefficient Functionals Over the Class of Functions Convex in the Direction of the Imaginary Axis

Cho, Nak Eun; Kowalczyk, Bogumiła; Lecko, Adam

doi:10.1017/s0004972718001429

Cited by 41 publications

(15 citation statements)

References 20 publications

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“…This was shown not to be the case even when n = 2 [5] and that the following sharp bounds hold. −1 ≤ |a 3 | − |a 2 | ≤ 3 4 + e −λ 0 (2e −λ 0 − 1) = 1.029 . .…”

Section: Introductionmentioning

confidence: 99%

On the Difference of Coefficients of Ozaki Close-to-Convex Functions

Sim

Thomas

2020

Bull. Aust. Math. Soc.

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Let $f$ be analytic in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ 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.

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“…This was shown not to be the case even when n = 2 [5] and that the following sharp bounds hold. −1 ≤ |a 3 | − |a 2 | ≤ 3 4 + e −λ 0 (2e −λ 0 − 1) = 1.029 . .…”

Section: Introductionmentioning

confidence: 99%

On the Difference of Coefficients of Ozaki Close-to-Convex Functions

Sim

Thomas

2020

Bull. Aust. Math. Soc.

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“…The formula (1.8) can be found in [28, p. 166]. The formula (1.9) was shown in a recent paper [7], where the extremal functions (1.11) and (1.12) were computed also. For c 1 ≥ 0 the formula (1.9) is due to by Libera and Zlotkiewicz [22,23].…”

Section: Introductionmentioning

confidence: 99%

“…Re e iβ f (z) g(z) > 0, z ∈ D. (1.2) in P. However both classes are rotation non-invariant. Thus to solve discussed problems we will apply a general formula for c 3 recently found in [7]. The formula (1.7) was proved by Carathéodory [3] (see e.g., [10, p. 41]).…”

Section: Introductionmentioning

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, a great many papers have been devoted to the estimation of determinants whose entries are coefficients of functions in A or its subclasses. Hankel matrices, i.e., square matrices which have constant entries along the reverse diagonal and the generalized Zalcman functional J m,n ( f ) := a m+n−1 − a m a n , m, n ∈ N, are of particular interest (see, e.g., [5,6,8,13,15,16,[18][19][20]25]). Also of interest are the determinants of symmetric Toeplitz matrices, the study of which was initiated in [1].…”

Section: Introduction and Definitionsmentioning

confidence: 99%