Ma-Minda φ-classes studied in this paper:
Abstract
The objective of this paper is to investigate the bounds of third and
fourth Hankel determinants for a generalized subclass of bounded turning func-
tions associated with sine function, in the open unit disc E = {z ∈C : |z| < 1}.
The results are also extended to two-fold and three-fold symmetric functions.
This investigation will generalize the resuls of some earlier works.
Mathematics Subject Classification (2010): 30C45, 30C50, 30C80.
Results & Lemmas (13)
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 1.2.
Lemma 1.2. [3] If p ∈P, then |pk| ≤2, k ∈N, p2 −p2 1 2 ≤2 −|p1|2 2, |pi+j −µpipj| ≤2, 0 ≤µ ≤1, |pn+2k −λpnp2 k| ≤2(1 + 2λ), (λ ∈R), |pmpn…
Lemma 1.2. [3] If p ∈P, then |pk| ≤2, k ∈N, p2 −p2 1 2 ≤2 −|p1|2 2 , |pi+j −µpipj| ≤2, 0 ≤µ ≤1, |pn+2k −λpnp2 k| ≤2(1 + 2λ), (λ ∈R), |pmpn −pkpl| ≤4, (m + n = k + l; m, n ∈N), and for complex number ρ, we have |p2 −ρp2
Lemma 1.3.
Lemma 1.3. [3] Let p ∈P, then |Jp3 1 −Kp1p2 + Lp3| ≤2|J| + 2|K −2J| + 2|J −K + L|. In particular, it is proved in [22] that |p3 1 −2p1p2 +…
Lemma 1.3. [3] Let p ∈P, then |Jp3 1 −Kp1p2 + Lp3| ≤2|J| + 2|K −2J| + 2|J −K + L|. In particular, it is proved in [22] that |p3 1 −2p1p2 + p3| ≤2.
Lemma 1.4.
Lemma 1.4. [13, 14] If p ∈P, then 2p2 = p2 1 + (4 −p2 1)x, 4p3 = p3 1 + 2p1(4 −p2 1)x −p1(4 −p2 1)x2 + 2(4 −p2 1)(1 −|x|2)z, for |x| ≤1 and…
Lemma 1.4. [13, 14] If p ∈P, then 2p2 = p2 1 + (4 −p2 1)x, 4p3 = p3 1 + 2p1(4 −p2 1)x −p1(4 −p2 1)x2 + 2(4 −p2 1)(1 −|x|2)z, for |x| ≤1 and |z| ≤1.
Lemma 1.5.
Lemma 1.5. [23] Let m, n, l and r satisfy the inequalities 0 < m < 1, 0 < r < 1 and 8r(1−r) (mn −2l)2 + (m(r + m) −n)2 +m(1−m)(n−2rm)2…
Lemma 1.5. [23] Let m, n, l and r satisfy the inequalities 0 < m < 1, 0 < r < 1 and 8r(1−r) (mn −2l)2 + (m(r + m) −n)2 +m(1−m)(n−2rm)2 ≤4m2(1−m)2r(1−r). If p ∈P, then lp4 1 + rp2 2 + 2mp1p3 −3 2np2 1p2 −p4 ≤2.
Theorem 2.1.
Theorem 2.1. If f ∈Rα sin, then |a2| ≤ 1 1 + α, (2.1) |a3| ≤ 1 1 + 2α, (2.2) |a4| ≤ 1 1 + 3α, (2.3) and
Theorem 2.1. If f ∈Rα sin, then |a2| ≤ 1 1 + α, (2.1) |a3| ≤ 1 1 + 2α, (2.2) |a4| ≤ 1 1 + 3α, (2.3) and
Theorem 2.4.
Theorem 2.4. If f ∈Rα sin and µ is any complex number, then |a3 −µa2 2| ≤ 1 1 + 2α max 1, (1 + 2α) (1 + α)2 |µ| . (2.15)
Theorem 2.4. If f ∈Rα sin and µ is any complex number, then |a3 −µa2 2| ≤ 1 1 + 2α max 1, (1 + 2α) (1 + α)2 |µ| . (2.15)
Theorem 2.8.
Theorem 2.8. If f ∈Rα sin, then |a2a3 −a4| ≤ 1 1 + 3α. (2.18)
Theorem 2.8. If f ∈Rα sin, then |a2a3 −a4| ≤ 1 1 + 3α. (2.18)
Theorem 2.11.
Theorem 2.11. If f ∈Rα sin, then |a2a4 −a2 3| ≤ 1 (1 + 2α)2. (2.20)
Theorem 2.11. If f ∈Rα sin, then |a2a4 −a2 3| ≤ 1 (1 + 2α)2 . (2.20)
Theorem 2.14.
Theorem 2.14. If f ∈Rα sin, then |H3(1)| ≤(2 + 8α + 4α2)(1 + 3α)2 + (1 + 4α)(1 + 2α)3 (1 + 2α)3(1 + 3α)2(1 + 4α). (2.21)
Theorem 2.14. If f ∈Rα sin, then |H3(1)| ≤(2 + 8α + 4α2)(1 + 3α)2 + (1 + 4α)(1 + 2α)3 (1 + 2α)3(1 + 3α)2(1 + 4α) . (2.21)
Theorem 2.17.
Theorem 2.17. If f ∈Rα sin, then |H4(1)|≤ 2 (1 + 2α)2(1 + 4α) 1 + 4α + 2α2 (1 + 2α)(1 + 6α) + 3 + 12α + 2α2 (1 + 4α)2 + 2 + 8α + 4α2 (1 +…
Theorem 2.17. If f ∈Rα sin, then |H4(1)|≤ 2 (1 + 2α)2(1 + 4α) 1 + 4α + 2α2 (1 + 2α)(1 + 6α) + 3 + 12α + 2α2 (1 + 4α)2 + 2 + 8α + 4α2 (1 + 3α)(1 + 5α) + 1
Theorem 2.1
Theorem 2.1, Theorem 2.4, Theorem 2.8, Theorem 2.11 and Theorem 2.14, the proof of the Theorem 2.17 is obvious. □ For α = 0, Theorem 2.17…
Theorem 2.1, Theorem 2.4, Theorem 2.8, Theorem 2.11 and Theorem 2.14, the proof of the Theorem 2.17 is obvious. □ For α = 0, Theorem 2.17 yields the following result:
Theorem 3.1.
Theorem 3.1. If f ∈Rα(2) sin, then |H3(1)| ≤ 1 (1 + 2α)(1 + 4α). (3.3)
Theorem 3.1. If f ∈Rα(2) sin , then |H3(1)| ≤ 1 (1 + 2α)(1 + 4α). (3.3)
Theorem 3.4.
Theorem 3.4. If f ∈Rα(3) sin, then |H3(1)| ≤ 1 (1 + 3α)2. (3.9)
Theorem 3.4. If f ∈Rα(3) sin , then |H3(1)| ≤ 1 (1 + 3α)2 . (3.9)
Function classes studied:
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