The Sharp Bound for the Hankel Determinant of the Third Kind for Convex Functions

Kowalczyk, Bogumiła; Lecko, Adam; Sim, Young Jae

doi:10.1017/s0004972717001125

Cited by 142 publications

(83 citation statements)

References 16 publications

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“…Taking modulus on either side of each expression in (17) and applying Lemma 3 and Lemma 5, we obtain…”

Section: Mains Resultsmentioning

confidence: 99%

“…Theorem 9 If f ∈ S * s (e z ) then |a 2 a 4 − a 2 3 | ≤ 3 8 . Proof: From equation (17) of Theorem 6, we have…”

Section: Mains Resultsmentioning

confidence: 99%

“…Babalola [4] was the first person to study the upper bound of H 3 (1) for subclasses of S. Interested readers can see the work carried by several researchers like Vamshee Krishna et al ( [45], [46]), Prajapat et al ( [32], [33]),Altinkaya and Yalcin [3],Cho et al [8], lecko et al [19], Kowalczyk et al [17],Mohd Narzan et al [27]. Mendiratta et al [26] introduced and studied the class of starlike functions S * e = S * (e z ) defined by…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Third Hankel Determinant for A Class of Functions with Respect to Symmetric Points Associated with Exponential Function

Ganesh¹,

Sharma

Laxmi³

2020

Wseas Transactions on Mathematics

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“…Taking modulus on either side of each expression in (17) and applying Lemma 3 and Lemma 5, we obtain…”

Section: Mains Resultsmentioning

confidence: 99%

“…Theorem 9 If f ∈ S * s (e z ) then |a 2 a 4 − a 2 3 | ≤ 3 8 . Proof: From equation (17) of Theorem 6, we have…”

Section: Mains Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Third Hankel Determinant for A Class of Functions with Respect to Symmetric Points Associated with Exponential Function

Ganesh¹,

Sharma

Laxmi³

2020

Wseas Transactions on Mathematics

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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 H 3,1 ( f ) over several subfamilies of A were studied in [8][9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients

Kwon¹,

Sim²

2019

Preprint

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Let ${\mathcal{SR}}^*$ 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 $z\in\mathbb{D}:=\{z\in\mathbb{C}:|z|<1 \}$ and $a_n:=f^{(n)}(0)/n!$ is real for all $n\in\mathbb{N}$. In the present paper, the sharp estimates of the third Hankel determinant $H_{3,1}$ over the class ${\mathcal{SR}}^*$ are computed.

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The Sharp Bound for the Hankel Determinant of the Third Kind for Convex Functions

Cited by 142 publications

References 16 publications

Third Hankel Determinant for A Class of Functions with Respect to Symmetric Points Associated with Exponential Function

Third Hankel Determinant for A Class of Functions with Respect to Symmetric Points Associated with Exponential Function

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

The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients

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