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

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

Section: Zalcman Functional and Hankel Determinantmentioning

confidence: 99%

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

Sim

2019

Results Math

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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%

Sharp Bounds of the Hermitian Toeplitz Determinants for Some Classes of Close-to-Convex Functions

Bull. Malays. Math. Sci. Soc.

Śmiarowska

2021

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“…( of A has been intensively studied. Many authors have examined the second Hankel determinant H 2,2 ( f ) = a 2 a 4 − a 2 3 (see, for example, [2,4,5,10,11,17,21]). We investigate H 2,2 ( f ) and also the functional J 2,3 ( f ) := a 2 a 3 − a 4 , a specific case of the generalised Zalcman functional J n,m ( f ) := a n a m − a n+m−1 for n, m ∈ N \ {1}, which was investigated by Ma [20] (see also [23] for other results).…”

Section: Introductionmentioning

confidence: 99%

“…We investigate H 2,2 ( f ) and also the functional J 2,3 ( f ) := a 2 a 3 − a 4 , a specific case of the generalised Zalcman functional J n,m ( f ) := a n a m − a n+m−1 for n, m ∈ N \ {1}, which was investigated by Ma [20] (see also [23] for other results). Many authors (see, for example, [1,2,4,5,11,14]) have computed upper bounds for the functional J 2,3 over various subclasses of A.…”

Section: Introductionmentioning

confidence: 99%

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

Cho

Kowalczyk

2019

Bull. Aust. Math. Soc.

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We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant $H_{2,2}$ and Zalcman functional $J_{2,3}$ over the class ${\mathcal{C}}{\mathcal{V}}$ of analytic functions $f$ normalised such that $\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$ for $z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, that is, the subclass of the class of functions convex in the direction of the imaginary axis.