Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
In this paper, we have obtained upper bound on third Hankel
determinant for the functions belonging to the class of close-to-convex
functions.
1
Results & Lemmas (11)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1
Lemma 1 (Carath´eodory’s Lemma [4], see also [5, p. 41]). Let the function p ∈P be given by the series then the sharp estimate |cn| ≤2, n =…
Lemma 1 (Carath´eodory’s Lemma [4], see also [5, p. 41]). Let the function p ∈P be given by the series then the sharp estimate |cn| ≤2, n = 1, 2, · · · holds. The inequality is sharp for each n.
Lemma 2
Lemma 2 (cf. [14, p. 254], see also [15]). Let the function p ∈P be given by (2), then 2c2 = c2 1 + x(4 −c2 1) for some x, |x| ≤1, and 4c3…
Lemma 2 (cf. [14, p. 254], see also [15]). Let the function p ∈P be given by (2), then 2c2 = c2 1 + x(4 −c2 1) for some x, |x| ≤1, and 4c3 = c3 1 + 2c1(4 −c2 1)x −c1(4 −c2 1)x2 + 2(4 −c2 1)(1 −|x|2)z for some z, |z| ≤1.
Lemma 3
Lemma 3 ([5, p. 44]). If f ∈S∗ be given by (1), then |an| ≤n (n = 2, 3,... ). Strict inequality holds for all n unless f is rotation of the…
Lemma 3 ([5, p. 44]). If f ∈S∗ be given by (1), then |an| ≤n (n = 2, 3, . . . ). Strict inequality holds for all n unless f is rotation of the Koebe function k(z) = z/(1 −z)2.
Lemma 4
Lemma 4 ([23]). If f ∈C be given by (1), then |an| ≤n (n = 2, 3,... ). Equality holds for all n when f is rotation of the Koebe function.
Lemma 4 ([23]). If f ∈C be given by (1), then |an| ≤n (n = 2, 3, . . . ). Equality holds for all n when f is rotation of the Koebe function.
Lemma 5
Lemma 5 ([10]). If f ∈S∗be given by (1), then for any real number μ, we have |a3 −μa2 2| ≤ ⎧ ⎨ ⎩ 3 −4μ, if μ ≤1 2 1, if 1 2 ≤μ ≤1
Lemma 5 ([10]). If f ∈S∗be given by (1), then for any real number μ, we have |a3 −μa2 2| ≤ ⎧ ⎨ ⎩ 3 −4μ, if μ ≤1 2 1, if 1 2 ≤μ ≤1
Lemma 6
Lemma 6 ([11]). If f ∈C be given by (1), then |a3 −a2 2| ≤1. There is a function in C such that equality holds.
Lemma 6 ([11]). If f ∈C be given by (1), then |a3 −a2 2| ≤1. There is a function in C such that equality holds.
Lemma 7
Lemma 7 ([12]). If f ∈S∗be given by (1), then |a2a4 −a2 3| ≤1. Equality is attended for the the Koebe function.
Lemma 7 ([12]). If f ∈S∗be given by (1), then |a2a4 −a2 3| ≤1. Equality is attended for the the Koebe function.
Lemma 8
Lemma 8 ([1]). If f ∈S∗be given by (1), then |a2a3 −a4| ≤2. Equality is attained by Koebe function. 2 Main results Our first main result is…
Lemma 8 ([1]). If f ∈S∗be given by (1), then |a2a3 −a4| ≤2. Equality is attained by Koebe function. 2 Main results Our first main result is contained in the following theorem:
Theorem 1
Theorem 1 Let the function f ∈C be given by (1), then |a2a3 −a4| ≤3. (6)
Theorem 1 Let the function f ∈C be given by (1), then |a2a3 −a4| ≤3. (6)
Theorem 2
Theorem 2 Let the function f ∈C be given by (1), then H2(2) = |a2a4 −a2 3| ≤85 36. (14)
Theorem 2 Let the function f ∈C be given by (1), then H2(2) = |a2a4 −a2 3| ≤85 36. (14)
Theorem 3
Theorem 3 Let the function f ∈C be given by (1), then |H3(1)| ≤ 289 12. (18)
Theorem 3 Let the function f ∈C be given by (1), then |H3(1)| ≤ 289 12 . (18)
Function classes studied:
Related Papers