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Abstract
The aim of this paper is to obtain an upper bound to the second Hankel the determinant for starlike and convex functions of order.
Results & Lemmas (4)
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Lemma 1.3.
Lemma 1.3. ([1], [2])If c ∈P, then |cn| ≤2, for each n ≥1.
Lemma 1.3. ([1], [2])If c ∈P, then |cn| ≤2, for each n ≥1.
Lemma 1.4.
Lemma 1.4. ([3], [4], [5])If c ∈P, then 2c2 = c2 1 + (4 −c2 1)x, 4c3 = c3 1 + 2c1(4 −c2 1)x −c1(4 −c2 1)x2 + 2(4 −c2 1)(1 −|x|2)z, for some…
Lemma 1.4. ([3], [4], [5])If c ∈P, then 2c2 = c2 1 + (4 −c2 1)x, 4c3 = c3 1 + 2c1(4 −c2 1)x −c1(4 −c2 1)x2 + 2(4 −c2 1)(1 −|x|2)z, for some x and z satisfying |x| ≤1, |z| ≤1 and c1 ∈[0, 2]. We employ techniques similar to these used earlier by Amourah et al. ([15], [16], [17], [18], [20]) and Al-Hawary et al. [19]. 2. Main Result
Theorem 2.1.
Theorem 2.1. If f(z) ∈S∗ α, then a2a4 −a2 3 ≤(2 −α 2)2 3 h 4 (2 −α 2 )2 −1 + 3 i
Theorem 2.1. If f(z) ∈S∗ α, then a2a4 −a2 3 ≤(2 −α 2)2 3 h 4 (2 −α 2 )2 −1 + 3 i
Theorem 2.3.
Theorem 2.3. If f(z) ∈Cα, then a2a4 −a2 3 ≤(2 −α 2)2 144 17(2 −α 2 )2 + 2(2 −α 2 ) + 17 1 + (2 −α 2)2 , (1 ≤α ≤2, z ∈U). (2.12)
Theorem 2.3. If f(z) ∈Cα, then a2a4 −a2 3 ≤(2 −α 2)2 144 17(2 −α 2 )2 + 2(2 −α 2 ) + 17 1 + (2 −α 2)2 , (1 ≤α ≤2, z ∈U). (2.12)
Function classes studied:
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