Abstract
The purpose of the present paper is to introduce a new subclasses of the function class of bi-univalent functions defined in the open unit disc. Furthermore, we obtain estimates on the coefficients $|a_{2}|$ and $|a_{3}|$ for functions of this class. Some results related to this work will be briefly indicated.
Results & Lemmas (4)
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Lemma 1.1. · coeff
Lemma 1.1. If h ∈p then |ck| < 1, for each k, where p is the family of all functions h analytic in U for which ℜ h(z) > 0, then h(z) = 1 +…
Lemma 1.1. If h ∈p then |ck| < 1, for each k, where p is the family of all functions h analytic in U for which ℜ{h(z)} > 0, then h(z) = 1 + c1z + c2z2 + c3z3 + ... , z ∈U. 2. COEFFICIENT BOUNDS FOR THE FUNCTION CLASS HΣ(ψ) In the sequel, it is assumed that ϕ . is an analytic function with positive real part in the unit disk D, satisfying ψ(0) = 1, ψ ′(0) > 0, and ψ(D) is symmetric with respect to the real axis. Such a function has a Taylor series of the form ψ(z) = 1 + B1z + B2z2 + B3z3 + ....,
Theorem 2.2.
Theorem 2.2. If f given by (1) is in the class HΣ(k, ψ), then |a2| ≤
Theorem 2.2. If f given by (1) is in the class HΣ(k, ψ), then |a2| ≤
Theorem 3.2.
Theorem 3.2. Let the function f(z) given by (1) be in the class QΣ(α, µ, λ), n ∈N0, 0 ≤β < 1, λ ≥1. Then |a2| ≤2α
Theorem 3.2. Let the function f(z) given by (1) be in the class QΣ(α, µ, λ), n ∈N0, 0 ≤β < 1, λ ≥1. Then |a2| ≤2α
Theorem 4.2.
Theorem 4.2.. Let f(z) given by (1) be in the class HΣ(β, µ, λ), 0 ≤β < 1, µ ≥0, and λ ≥1. Then |a2| ≤ k + α −1 k −2 k + α…
Theorem 4.2. . Let f(z) given by (1) be in the class HΣ(β, µ, λ), 0 ≤β < 1, µ ≥0, and λ ≥1. Then |a2| ≤ k + α −1 k −2 k + α −2 k −2
Function classes studied:
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