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Abstract
In the present investigation we introduce some subclasses of the function class Σ of bi-univalent functions defined in the open unit disk U,
which are associated with the quasi-subordination. We obtain the estimates on initial coefficients |a2| and |a3| for the functions in these
subclasses. Also several related subclasses are considered and connection with some known results are established.
Results & Lemmas (7)
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 2.2
Theorem 2.2: Let f(z) given by (1) be in the class Rq Σ(λ,φ). Then, |a2| ≤min ( |A0|B1 1+λ, r |A0|(B1 +|B2 −B1|) 1+2λ ) (6) and |a3| ≤min (…
Theorem 2.2: Let f(z) given by (1) be in the class Rq Σ(λ,φ). Then, |a2| ≤min ( |A0|B1 1+λ , r |A0|(B1 +|B2 −B1|) 1+2λ ) (6) and |a3| ≤min ( (|A0|+|A1|)B1
Corollary 2.3
Corollary 2.3: Let the function f(z) given by (1) be in the class RΣ(λ,φ). Then, |a2| ≤min ( B1 1+λ, r B1 +|B2 −B1| 1+2λ )
Corollary 2.3: Let the function f(z) given by (1) be in the class RΣ(λ,φ). Then, |a2| ≤min ( B1 1+λ , r B1 +|B2 −B1| 1+2λ )
Corollary 2.4
Corollary 2.4: Let the function f(z) given by (1) be in the class RΣ(φ). Then, |a2 ≤min ( B1 2, r B1 +|B2 −B1| 3 ) and |a3| ≤min ( B1
Corollary 2.4: Let the function f(z) given by (1) be in the class RΣ(φ). Then, |a2 ≤min ( B1 2 , r B1 +|B2 −B1| 3 ) and |a3| ≤min ( B1
Theorem 2.1
Theorem 2.1 given by Ali et al. [1], respectively.
Theorem 2.1 given by Ali et al. [1], respectively.
Theorem 2
Theorem 2 given by Srivastava et al. [16], respectively. 3. Coefficient Estimates for the Function Class S ∗,q Σ (φ) Definition 3.1: A…
Theorem 2 given by Srivastava et al. [16], respectively. 3. Coefficient Estimates for the Function Class S ∗,q Σ (φ) Definition 3.1: A function f ∈Σ given by (1) is said to be in the class S ∗,q Σ (φ) if the following quasi-subordination holds: " z f ′(z) f(z) −1 # ≺q (φ(z)−1) and "
Theorem 3.2
Theorem 3.2: Let f(z) given by (1) be in the class S ∗,q Σ (φ). Then, |a2| ≤min L,M,N (26) where, L = p |A0|(B1 +|B2 −B1|), M = q A2 0B2…
Theorem 3.2: Let f(z) given by (1) be in the class S ∗,q Σ (φ). Then, |a2| ≤min{L,M,N} (26) where, L = p |A0|(B1 +|B2 −B1|), M = q A2 0B2 1+|A0|(B1+|B2−B1|) 2 , N =
Theorem 4.2
Theorem 4.2: Let f(z) given by (1) be in the class K q Σ (φ). Then, |a2| ≤min s A2 0B2 1 +|A0|(B1 +|B2 −B1|) 6, |A0|B1 2
Theorem 4.2: Let f(z) given by (1) be in the class K q Σ (φ). Then, |a2| ≤min s A2 0B2 1 +|A0|(B1 +|B2 −B1|) 6 , |A0|B1 2
Function classes studied:
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