Abstract
By making use of the linear operator Θλ,n
m , m ∈N = {1,2,3,...} and λ, n ∈N0 =
N∪{0} given by the authors, a class of analytic functions Sλ,n
m (α,σ)(|α| < π/2, 0 ≤σ < 1) is
introduced. The object of the present paper is to obtain sharp upper bound for functional
¯¯a2a4 −a2
3
¯¯.
Results & Lemmas (5)
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.
Theorem 1.1
Theorem 1.1 ([15]). Let f ∈S and−π/2 < α < π/2. Then f (z) is a spirallike function of type α on U if ℜ ½ eiα z f ′(z) f (z) ¾ > 0, z ∈U.…
Theorem 1.1 ([15]). Let f ∈S and−π/2 < α < π/2. Then f (z) is a spirallike function of type α on U if ℜ ½ eiα z f ′(z) f (z) ¾ > 0, z ∈U. We denoted this class by Sα. Corresponding author: Maslina Darus. 2010 Mathematics Subject Classification. 30C45. Key words and phrases. Hankel determinant, Positive real functions, Linear operator. 455
Lemma 1.2
Lemma 1.2 ([9]). Let the function p ∈P and be given by the series (1.5). Then, the sharp esti- mate |ck| ≤2 (k ∈N) holds.
Lemma 1.2 ([9]). Let the function p ∈P and be given by the series (1.5). Then, the sharp esti- mate |ck| ≤2 (k ∈N) holds.
Lemma 1.3
Lemma 1.3 ([7] and [8]). Let the function p ∈P be given by the series (1.5). Then 2c2 = c2 1 + x(4−c2 1) (1.7) for some x, |x| ≤1 and 4c3 =…
Lemma 1.3 ([7] and [8]). Let the function p ∈P be given by the series (1.5). Then 2c2 = c2 1 + x(4−c2 1) (1.7) for some x, |x| ≤1 and 4c3 = c3 1 +2(4−c2 1)c1x −c1(4−c2 1)x2 +2(4−c2 1)(1−|x|2)z (1.8) for some z, |z| ≤1.
Theorem 2.1.
Theorem 2.1. Let the function f given by (1.1) be in the class Sλ,n m (α,σ). Then ¯¯a2a4 −a2 3 ¯¯ ≤4m2(1−σ)2(1+m)2 cos2 α 32n(λ+1)2(λ+2)2.…
Theorem 2.1. Let the function f given by (1.1) be in the class Sλ,n m (α,σ). Then ¯¯a2a4 −a2 3 ¯¯ ≤4m2(1−σ)2(1+m)2 cos2 α 32n(λ+1)2(λ+2)2 . (2.1) The estimate (2.1) is sharp.
Corollary 2.1.
Corollary 2.1. If f ∈R then ¯¯a2a4 −a2 3 ¯¯ ≤4 9. The result is sharp. Acknowledgement The work here is fully supported by MOHE Grant:…
Corollary 2.1. If f ∈R then ¯¯a2a4 −a2 3 ¯¯ ≤4 9. The result is sharp. Acknowledgement The work here is fully supported by MOHE Grant: UKM-ST-06-FRGS0244-2010, Malaysia. References [1] A. Mohammed and M. Darus, An operator defined by convolution involving the generalized Hurwitz–Lerch zeta function , Submitted [2] A. Janteng, S. A. Halim, and M. Darus, Coefficient inequality for a function whose derivative has positive real part , J.Ineq. Pure and Appl. Math.,7(2)(2006), 1–5. [3] A. Janteng, S. A.
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