Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
By using Jackson q-derivative, some characterizations in terms of convolutions for two
classes of analytic functions in the open unit disc are given. Also, coefficient conditions
and inclusion properties for functions in these classes are found.
Results & Lemmas (9)
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.
Proposition 2
Proposition 2 If f ∈A, then and
Proposition 2 If f ∈A, then and
Theorem 3
Theorem 3 A function f of the form (1) in the class S∗,ζ(A, B) if and only if where C = Cθ = e−iθ+A A−B, θ ∈[0, 2π).
Theorem 3 A function f of the form (1) in the class S∗ ,ζ(A, B) if and only if where C = Cθ = e−iθ+A A−B , θ ∈[0, 2π).
Corollary 4
Corollary 4 A function f of the form (1) in the class S∗ ζ (A, B) if and only if where C = Cθ = e−iθ+A A−B, θ ∈[0, 2π).
Corollary 4 A function f of the form (1) in the class S∗ ζ (A, B) if and only if where C = Cθ = e−iθ+A A−B , θ ∈[0, 2π).
Corollary 6
Corollary 6 A function f of the form (1) is in the class Cζ(A, B) if and only if 1 z
Corollary 6 A function f of the form (1) is in the class Cζ(A, B) if and only if 1 z
Theorem 3
Repeated OCR label — verify against source
Theorem 3, we obtain the result obtained by Lashin [16]. 2 Letting ζ →1−1, A = [b(1 + m) −m] and B = −m with m = 1 −1 M and M > 1 2 in…
Theorem 3, we obtain the result obtained by Lashin [16]. 2 Letting ζ →1−1, A = [b(1 + m) −m] and B = −m with m = 1 −1 M and M > 1 2 in Corollaries 4 and 6, respectively, we obtain the results obtained by El-Ashwah [26,
Theorem 2.1
Theorem 2.1 and Theorem 2.4]. 3 Taking ζ →1−1, A = 1 −2α, B = −1 and eiθ = x(|x| = 1) in Corollaries 4 and 6, we obtain the results…
Theorem 2.1 and Theorem 2.4]. 3 Taking ζ →1−1, A = 1 −2α, B = −1 and eiθ = x(|x| = 1) in Corollaries 4 and 6, we obtain the results obtained by Silverman et al. [23, Theorems 1,2]. 4 Taking ζ →1−1and eiθ = x(|x| = 1) in Corollaries 4 and 6, respectively, we obtain the results obtained by Padmanabhan and Ganesan [27, Theorem 1,2].
Theorem 9
Theorem 9 A necessary and sufficient condition for the function f of the form (1) to be in the class S∗,ζ(n, A, B) is for all θ ∈[0, 2π)…
Theorem 9 A necessary and sufficient condition for the function f of the form (1) to be in the class S∗ ,ζ(n, A, B) is for all θ ∈[0, 2π) and z ∈U.
Theorem 10
Theorem 10 If the function f defined by (1) satisfies the inequality then f (z) ∈S∗,ζ(n, A, B).
Theorem 10 If the function f defined by (1) satisfies the inequality then f (z) ∈S∗ ,ζ(n, A, B).
Theorem 11
Theorem 11 S∗,ζ(n + 1, A, B) ⊂S∗,ζ(n, A, B).
Theorem 11 S∗ ,ζ(n + 1, A, B) ⊂S∗ ,ζ(n, A, B).
Definitions (1)
Def 1
Definition 1 For f ∈A, we say f belongs to class S∗,ζ(A, B), if and only if where 0 < ζ < 1, 0 ≤ ≤1, −1 ≤B < A ≤1, Dζ is Jackson…
Definition 1 For f ∈A , we say f belongs to class S∗ ,ζ(A, B) , if and only if where 0 < ζ < 1, 0 ≤ ≤1, −1 ≤B < A ≤1, Dζ is Jackson q-derivative with q = ζ and ≺ denotes the usual subordination (see [9–11]). It is noticed that, by giving specific values to A, B and we obtain the following important subclasses studied by various authors in earlier works: 1. S∗ 0,q(1 −2α, −1) ≡S∗ q(α) and S∗ 1,q(1 −2α, −1) ≡Cq(α) are, respectively, the classes
Function classes studied:
Related Papers