Abstract
In this paper, we introduce and investigate a new subclass of bi-
univalent functions ∑ of complex order defined in the open unit disk, which are
associated with hypergeometric functions and satisfy subordinate conditions.
Furthermore, we find estimates on the Taylor-Maclaurin coefficients
2
|
|
a
and
3
|
|
a
for functions in the new subclass. Several (known or new) consequences of
the results are also pointed out.
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.
Lemma 2.1
Lemma 2.1 (see [22]) If, h P ∈ then | | 2 kc ≤ for each k, where P is the family of all functions h, analytic in U, for which
Lemma 2.1 (see [22]) If , h P ∈ then | | 2 kc ≤ for each k, where P is the family of all functions h, analytic in U, for which
Theorem 2.2
Theorem 2.2 Let the function f (z) given by Eq. (1) be in the class, (,, ) l m G γ λ ϕ Σ Then
Theorem 2.2 Let the function f (z) given by Eq. (1) be in the class , ( , , ) l m G γ λ ϕ Σ Then
Corollary 2.3
Corollary 2.3 Let the function f (z) given by Eq. (1) be in the class, (, ). l m S γ ϕ Σ
Corollary 2.3 Let the function f (z) given by Eq. (1) be in the class , ( , ). l m S γ ϕ Σ
Corollary 2.4
Corollary 2.4 Let the function f (z) given by Eq. (1) be in the class *(,, ). S γ λ ϕ Σ
Corollary 2.4 Let the function f (z) given by Eq. (1) be in the class *( , , ). S γ λ ϕ Σ
Corollary 2.5
Corollary 2.5 Let the function f (z) given by Eq. (1) be in the class *(, ). S γ ϕ Σ
Corollary 2.5 Let the function f (z) given by Eq. (1) be in the class *( , ). S γ ϕ Σ
Corollary 3.1
Corollary 3.1 Let f (z) given by Eq. (1) be in the class *(,, ), S α λ ϕ Σ and 2 1 2 = 2; = 2 B B α α (0 < 1;0
Corollary 3.1 Let f (z) given by Eq. (1) be in the class *( , , ), S α λ ϕ Σ and 2 1 2 = 2 ; = 2 B B α α (0 < 1;0
Corollary 3.2
Corollary 3.2 Let f (z) given by Eq. (1) be in the class *(,, ), S β λ ϕ Σ
Corollary 3.2 Let f (z) given by Eq. (1) be in the class *( , , ), S β λ ϕ Σ
Definitions (1)
Def 1.3
Definition 1.3 A function f ∈Σ given by Eq. (1) is said to be in the class, (,, ) l m G γ λ ϕ Σ if the following conditions are satisfied:
Definition 1.3 A function f ∈Σ given by Eq. (1) is said to be in the class , ( , , ) l m G γ λ ϕ Σ if the following conditions are satisfied:
Function classes studied:
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