Abstract
In this paper, we considered a family of analytic and univalent
functions having positive real parts in the unit disk and defined by a q-difference
operator. The coefficients, the Fekete-Szeg¨o estimates and the second Hankel
determinant were established for the family of functions. Our family of func-
tions generalized some earlier known ones and by varying some parameters,
our results also generalized some known ones.
Results & Lemmas (8)
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 2.1.
Lemma 2.1. [14]. |pk| ≤2 (k ∈N).
Lemma 2.1. [14]. |pk| ≤2 (k ∈N).
Lemma 2.2.
Lemma 2.2. [28]. p2 −λp2 1 ≤2 max 1, 2λ −1
Lemma 2.2. [28]. p2 −λp2 1 ≤2 max 1, 2λ −1
Lemma 2.3.
Lemma 2.3. [20]. 2p2 = p2 1 + (4 −p2 1)x, 4p2 2 = p4 1 + 2(4 −p2 1)p2 1x + (4 −p2 1)2x2 4p3 = p3 1 + 2(4 −p2 1)p1x −(4 −p2 1)p1x2 + 2(4 −p2…
Lemma 2.3. [20]. 2p2 = p2 1 + (4 −p2 1)x, 4p2 2 = p4 1 + 2(4 −p2 1)p2 1x + (4 −p2 1)2x2 4p3 = p3 1 + 2(4 −p2 1)p1x −(4 −p2 1)p1x2 + 2(4 −p2 1)(1 −|x|2)z
Theorem 3.1.
Theorem 3.1. Let α ∈(−π, π] and β ∈[0, 1). If f(z) ∈Aq(α, β), then ak = 2(1 −α)pk−1 [k]qψk (ψk = 2 + (1 + eiα)[k −1]q, k = 2, 3,... ). (3.1)
Theorem 3.1. Let α ∈(−π, π] and β ∈[0, 1). If f(z) ∈Aq(α, β), then ak = 2(1 −α)pk−1 [k]qψk (ψk = 2 + (1 + eiα)[k −1]q, k = {2, 3, . . .}). (3.1)
Corollary 3.3.
Corollary 3.3. Let α ∈(−π, π] and β ∈[0, 1). If f(z) ∈Aq(α, β), then |ak| ≤4(1 −β) [k]q|ψk| (k = 2, 3,... ) (3.4) where |ψk| is as defined…
Corollary 3.3. Let α ∈(−π, π] and β ∈[0, 1). If f(z) ∈Aq(α, β), then |ak| ≤4(1 −β) [k]q|ψk| (k = {2, 3, . . .}) (3.4) where |ψk| is as defined in Remark 3.2. In [4], Ali et al. investigated the Fekete-Szeg¨o problem related to the tth-root transformation for some subfamilies of S. For a function f ∈S of the form (1.1), the tth-root transformation is defined by (see [14]) F(z) = tp f(zt) = z + 1 t a2zt+1 + 1 t a3 −t −1
Theorem 3.4.
Theorem 3.4. Let f(z) ∈Aq(α, β). If α ∈(−π, π], β ∈[0, 1) and ρ ∈R, then |d2t+1 −ρd2 t+1| ≤ 8(1−β) t[3]q|ψ3| for ρ ≤1 2 t[2]2 q|ψ2…
Theorem 3.4. Let f(z) ∈Aq(α, β). If α ∈(−π, π], β ∈[0, 1) and ρ ∈R, then |d2t+1 −ρd2 t+1| ≤ 8(1−β) t[3]q|ψ3| for ρ ≤1 2 t[2]2 q|ψ2 2|
Theorem 3.5.
Theorem 3.5. Let f(z) ∈Aq(α, β). If α ∈(−π, π], β ∈[0, 1) and ϱ ∈C, then |d2t+1 −ϱd2 t+1| ≤4(1 −β) t[3]q|ψ3| max 1,
Theorem 3.5. Let f(z) ∈Aq(α, β). If α ∈(−π, π], β ∈[0, 1) and ϱ ∈C, then |d2t+1 −ϱd2 t+1| ≤4(1 −β) t[3]q|ψ3| max 1,
Theorem 3.6
Theorem 3.6 (Hankel Determinant: H2(2)). Let f(z) ∈Aq(α, β). If β ∈ [0, 1), then |H2(2)| = |a2a4 −a3 2| ≤18(1 −β)2 L + 16(1 −β)2 M where L…
Theorem 3.6 (Hankel Determinant: H2(2)). Let f(z) ∈Aq(α, β). If β ∈ [0, 1), then |H2(2)| = |a2a4 −a3 2| ≤18(1 −β)2 L + 16(1 −β)2 M where L = [2]q[4]qψ2ψ4 = [2]q[4]q ψ2 ψ4 > 0 M =
Function classes studied:
Related Papers