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Abstract
By applying Opoola differential operator, in this article, two new subclasses Mµ,β
H,σ(m, ψ, k, τ)
and Mµ,β
H,σ(m, ξ, k, τ) of bi-univalent functions class H defined in ▽are introduced and investigated. The
estimates on the coefficients |l2| and |l3| for functions of the classes are also obtained.
Results & Lemmas (3)
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Lemma 1.
Lemma 1. [6] Suppose u(z) ∈P and z ∈▽, then |wk| ≤2 for each k, where P is the family of all function u analytic in ▽for which ℜ(u(z)) > 0,…
Lemma 1. [6] Suppose u(z) ∈P and z ∈▽, then |wk| ≤2 for each k, where P is the family of all function u analytic in ▽for which ℜ(u(z)) > 0, u(z) = 1 + w1z + w2z2 + · · · . Open J. Math. Anal. 2020, 4(2), 74-79; doi:10.30538/psrp-oma2020.0064 https://pisrt.org/psr-press/journals/oma
Theorem 1.
Theorem 1. Let ℑ(z) ∈G be in the class Mµ,β H (m, ψ, k, τ), 0 < ψ ≤1, σ ≥1, τ ≥0, z ∈△, b ∈△, 0 ≤µ ≤β, m ∈N0, then |l2| ≤ 2ψ s 2ψ[(1 −σ)(1…
Theorem 1. Let ℑ(z) ∈G be in the class Mµ,β H (m, ψ, k, τ), 0 < ψ ≤1, σ ≥1, τ ≥0, z ∈△, b ∈△, 0 ≤µ ≤β, m ∈N0 , then |l2| ≤ 2ψ s 2ψ[(1 −σ)(1 + τ(2 + µ −β))m + σ(1 + τ(2 + µ −β))m+1]− ψ(ψ −1)[(1 −σ)(1 + τ(1 + µ −β))m + σ(1 + τ(1 + µ −β))m+1]2 (7) and |l3| ≤ 2ψ [(1 −σ)(1 + τ(2 + µ −β))m + σ(1 + τ(2 + µ −β))m+1] +
Theorem 2.
Theorem 2. Let ℑ(z) ∈G be in the class Mµ,β H (m, ξ, k, τ), 0 ≤ξ < 1, σ ≥1, τ ≥0, z ∈△, b ∈△, 0 ≤µ ≤β, m ∈N0, then |l2| ≤ s 2(1 −ξ) [(1…
Theorem 2. Let ℑ(z) ∈G be in the class Mµ,β H (m, ξ, k, τ), 0 ≤ξ < 1, σ ≥1, τ ≥0, z ∈△, b ∈△, 0 ≤µ ≤β, m ∈N0, then |l2| ≤ s 2(1 −ξ) [(1 −σ)(1 + τ(2 + µ −β))m + σ(1 + τ(2 + µ −β))m+1], (22) and |l3| ≤ 4(1 −ξ)2 [(1 −σ)(1 + τ(2 + µ −β))m + σ(1 + τ(2 + µ −β))m+1]2 + 2(1 −ξ)
Function classes studied:
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