Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
In the present paper, we consider certain classes of bi-univalent Bazilevi˜c functions with bounded
boundary rotation involving S˘al˘agean operator to obtain the estimates of their second and third coefficients. Further,
certain special cases are also indicated. Some interesting remarks about the results presented here are also discussed.
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 1.1.
Lemma 1.1. [3, Theorem 5 with p = 1] If p(z) = 1 + ∞ P n=1 cnzn ∈Pλ k (α) in U, then (1.8) |cn| ≤(1 −α) k cos λ (n ∈N). The result is…
Lemma 1.1. [3, Theorem 5 with p = 1] If p(z) = 1 + ∞ P n=1 cnzn ∈Pλ k (α) in U, then (1.8) |cn| ≤(1 −α) k cos λ (n ∈N) . The result is sharp. Equality is attained for the odd coefficients and even coefficients, respectively, for the functions p1 (z) = 1 + (1 −α) cos λ e−iλ k + 2 4 1 −z
Theorem 2.1.
Theorem 2.1. Let f(z) given by (1.1) belongs to the class Bm Σ (γ, δ, b; k) with δ ̸= 1 − 3m 22m−1, δ ̸= −γ and δ ̸= −2γ, then (2.9) |a2|…
Theorem 2.1. Let f(z) given by (1.1) belongs to the class Bm Σ (γ, δ, b; k) with δ ̸= 1 − 3m 22m−1 , δ ̸= −γ and δ ̸= −2γ, then (2.9) |a2| ≤min (s |b| k |(δ −1) 22m−1 + 3m| |δ + 2γ|, |b| k 2m |δ + γ| ) and (2.10)
Corollary 2.1.
Corollary 2.1. Let f(z) given by (1.1) belongs to the class BΣ (γ, δ; k) with δ ̸= −1, δ ̸= −γ and δ ̸= −2γ, then |a2| ≤min (s 2k |δ + 1|…
Corollary 2.1. Let f(z) given by (1.1) belongs to the class BΣ (γ, δ; k) with δ ̸= −1, δ ̸= −γ and δ ̸= −2γ, then |a2| ≤min (s 2k |δ + 1| |δ + 2γ|, k |δ + γ| )
Corollary 2.2.
Corollary 2.2. Let f(z) given by (1.1) belongs to the class BΣ (γ, δ, η) with 0 ≤η < 1, δ ̸= −1, δ ̸= −γ and δ ̸= −2γ, then |a2| ≤min (s 4…
Corollary 2.2. Let f(z) given by (1.1) belongs to the class BΣ (γ, δ, η) with 0 ≤η < 1, δ ̸= −1, δ ̸= −γ and δ ̸= −2γ, then |a2| ≤min (s 4 (1 −η) |δ + 1| |δ + 2γ|, 2 (1 −η) |δ + γ| ) and |a3| ≤2 (1 −η) |δ + 2γ| min ( 1 + 2 |δ + 1|; 1 + |δ + 2γ| |1 −δ| (1 −η)
Corollary 2.3.
Corollary 2.3. Let f(z) given by (1.1) belongs to the class BΣ (γ, η) with 0 ≤η < 1, γ ̸= −1 and γ ̸= −1 2, then |a2| ≤min (s 2 (1 −η) |2γ…
Corollary 2.3. Let f(z) given by (1.1) belongs to the class BΣ (γ, η) with 0 ≤η < 1, γ ̸= −1 and γ ̸= −1 2, then |a2| ≤min (s 2 (1 −η) |2γ + 1| , 2 (1 −η) |γ + 1| ) and |a3| ≤2 (1 −η) |2γ + 1| min ( 2, 1 + 4 |2γ + 1| (1 −η) |γ + 1|2
Corollary 2.4.
Corollary 2.4. Let f(z) given by (1.1) belongs to the class BΣ (δ, η) with δ ̸= −1 and δ ̸= −2, then |a2| ≤min (s 4 (1 −η) |δ + 1| |δ + 2|,…
Corollary 2.4. Let f(z) given by (1.1) belongs to the class BΣ (δ, η) with δ ̸= −1 and δ ̸= −2, then |a2| ≤min (s 4 (1 −η) |δ + 1| |δ + 2|, 2 (1 −η) |δ + 1| ) and |a3| ≤2 (1 −η) |δ + 2| min ( 1 + 2 |δ + 1|; 1 + |δ + 2| |1 −δ| (1 −η) |δ + 1|2
Corollary 2.5.
Corollary 2.5. Let f(z) given by (1.1) belongs to the class SΣ (b), then |a2| ≤min np 2 |b|, 2 |b| o and |a3| ≤|b| min 3, 1 + 2 |b|. Taking…
Corollary 2.5. Let f(z) given by (1.1) belongs to the class SΣ (b), then |a2| ≤min np 2 |b|, 2 |b| o and |a3| ≤|b| min {3, 1 + 2 |b|} . Taking δ = 0, m = 1, γ = 1 and k = 2 in Theorem 2.1, we obtain the following result for the functions belonging to the class CΣ (b).
Corollary 2.6.
Corollary 2.6. Let f(z) given by (1.1) belongs to the class CΣ (b), then |a2| ≤min np |b|, |b| o and |a3| ≤|b| 3 min 4, 1 + 2 |b|. Taking m…
Corollary 2.6. Let f(z) given by (1.1) belongs to the class CΣ (b), then |a2| ≤min np |b|, |b| o and |a3| ≤|b| 3 min {4, 1 + 2 |b|} . Taking m = 0 and b = (1 −α) e−iλ cos λ |λ| < π 2 , 0 ≤α < 1 in Theorem 2.1, we obtain the following result for the functions belonging to the class BΣ (γ, δ, α, λ; k).
Corollary 2.7.
Corollary 2.7. Let f(z) given by (1.1) belongs to the class BΣ (γ, δ, α, λ; k) with δ ̸= −1, δ ̸= −γ and δ ̸= −2γ, then |a2| ≤min (s 2k (1…
Corollary 2.7. Let f(z) given by (1.1) belongs to the class BΣ (γ, δ, α, λ; k) with δ ̸= −1, δ ̸= −γ and δ ̸= −2γ, then |a2| ≤min (s 2k (1 −α) cos λ |δ + 1| |δ + 2γ| , k (1 −α) cos λ |δ + γ| ) and |a3| ≤k (1 −α) cos λ |δ + 2γ| min ( 1 + 2
Corollary 2.8.
Corollary 2.8. Let f(z) given by (1.1) belongs to the class Sλ Σ (b), then |a2| ≤min np 2 |b| cos λ, 2 |b| cos λ o and |a3| ≤|b| cos λ min…
Corollary 2.8. Let f(z) given by (1.1) belongs to the class Sλ Σ (b), then |a2| ≤min np 2 |b| cos λ, 2 |b| cos λ o and |a3| ≤|b| cos λ min {3, 1 + 2 |b| cos λ} . Taking m = γ = 1, δ = 0, k = 2 and b →be−iλ cos λ |λ| < π 2 , 0 ≤α < 1 in Theorem 2.1, we obtain the following result for the functions belonging to the class Cλ
Corollary 2.9.
Corollary 2.9. Let f(z) given by (1.1) belongs to the class Cλ Σ (b), then |a2| ≤min np |b| cos λ, |b| cos λ o and |a3| ≤|b| cos λ 3 min 4,…
Corollary 2.9. Let f(z) given by (1.1) belongs to the class Cλ Σ (b), then |a2| ≤min np |b| cos λ, |b| cos λ o and |a3| ≤|b| cos λ 3 min {4, 1 + 2 |b| cos λ} . Taking δ = m = 0, γ = 1 and b = (1 −α) e−iλ cos λ |λ| < π 2 , 0 ≤α < 1
Corollary 2.10.
Corollary 2.10. Let f(z) given by (1.1) belongs to the class Sλ α (k) |λ| < π 2, 0 ≤α < 1 , then |a2| ≤min np k (1 −α) cos λ, k (1 −α)…
Corollary 2.10. Let f(z) given by (1.1) belongs to the class Sλ α (k) |λ| < π 2 , 0 ≤α < 1 , then |a2| ≤min np k (1 −α) cos λ, k (1 −α) cos λ o and |a3| ≤k (1 −α) cos λ 2 min {3, 1 + k (1 −α) cos λ} .
Corollary 2.11.
Corollary 2.11. Let f(z) given by (1.1) belongs to the class Cλ α (k) |λ| < π 2, 0 ≤α < 1 , then |a2| ≤min (r k (1 −α) cos λ 2, k (1 −α)…
Corollary 2.11. Let f(z) given by (1.1) belongs to the class Cλ α (k) |λ| < π 2 , 0 ≤α < 1 , then |a2| ≤min (r k (1 −α) cos λ 2 , k (1 −α) cos λ 2 ) and
Function classes studied:
Related Papers