Abstract
Inspired by the recent works of Srivastava et al. (HMS-AKM-PG), Frasin and Aouf (BAF-MKA) and others (Ali-Ravi-Ma-Mina-class,Caglar-Orhan,Goyal-Goswami,Xu-HMS-AML,Xu-HMS-AMC), we propose to investigate the coefficient estimates for a general class of analytic and bi-univalent functions. Also, we obtain estimates on the coefficients |a2| and |a3| for functions in this new class. Some interesting remarks, corollaries and applications of the results presented here are also discussed.
Results & Lemmas (9)
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Lemma 1.8.
Lemma 1.8. [12, 16] Let the function ϕ(z) given by ϕ(z) = ∞ X n=1 Bnzn, z ∈U be convex in U. Suppose also that the function h(z) given by…
Lemma 1.8. [12, 16] Let the function ϕ(z) given by ϕ(z) = ∞ X n=1 Bnzn, z ∈U be convex in U. Suppose also that the function h(z) given by h(z) = ∞ X n=1 hnzn, z ∈U is holomorphic in U. If h(z) ≺ϕ(z), z ∈U, then |hn| ≤|B1|, n ∈N = {1, 2, 3, . . .}.
Theorem 2.1.
Theorem 2.1. Let f(z) given by (1.1) be in the class N Pµ,λ Σ (β, h), 0 ≤α < 1, λ ≥1 and µgeq0, then |a2| ≤ s 2|B1| cos β (1 + µ)(2λ + µ)…
Theorem 2.1. Let f(z) given by (1.1) be in the class N Pµ,λ Σ (β, h), 0 ≤α < 1, λ ≥1 and µgeq0, then |a2| ≤ s 2|B1| cos β (1 + µ)(2λ + µ) (2.1) and |a3| ≤ 2|B1|cosβ (2λ + µ)(1 + µ), (2.2) where β ∈(−π/2, π/2).
Corollary 3.1.
Corollary 3.1. Let f(z) given by (1.1) be in the class N Pµ,λ Σ (β, 1+Az 1+Bz), then |a2| ≤ s 2(A −B) cos β (1 + µ)(2λ + µ) (3.1) and |a3|…
Corollary 3.1. Let f(z) given by (1.1) be in the class N Pµ,λ Σ (β, 1+Az 1+Bz), then |a2| ≤ s 2(A −B) cos β (1 + µ)(2λ + µ) (3.1) and |a3| ≤2(A −B)cosβ (2λ + µ)(1 + µ), (3.2) where β ∈(−π/2, π/2), µ ≥0 and λ ≥1.
Corollary 3.2.
Corollary 3.2. Let f(z) given by (1.1) be in the class N Pµ,λ Σ (β, α), 0 ≤α < 1, µ ≥0 and λ ≥1, then |a2| ≤ s 4(1 −α) cos β (1 + µ)(2λ +…
Corollary 3.2. Let f(z) given by (1.1) be in the class N Pµ,λ Σ (β, α), 0 ≤α < 1, µ ≥0 and λ ≥1, then |a2| ≤ s 4(1 −α) cos β (1 + µ)(2λ + µ) (3.3) and |a3| ≤ 4(1 −α)cosβ (2λ + µ)(1 + µ), (3.4) where β ∈(−π/2, π/2).
Corollary 3.4.
Corollary 3.4. Let f(z) given by (1.1) be in the class N Pµ,1 Σ (β, α), 0 ≤α < 1 and µ ≥0, then |a2| ≤ s 4(1 −α) cos β (1 + µ)(2 + µ) (3.5)…
Corollary 3.4. Let f(z) given by (1.1) be in the class N Pµ,1 Σ (β, α), 0 ≤α < 1 and µ ≥0, then |a2| ≤ s 4(1 −α) cos β (1 + µ)(2 + µ) (3.5) and |a3| ≤4(1 −α)cosβ (2 + µ)(1 + µ), (3.6) where β ∈(−π/2, π/2).
Corollary 3.5.
Corollary 3.5. Let f(z) given by (1.1) be in the class N P0,1 Σ (β, α), 0 ≤α < 1, then |a2| ≤ p 2(1 −α) cos β (3.7) and |a3| ≤2(1 −α)cosβ,…
Corollary 3.5. Let f(z) given by (1.1) be in the class N P0,1 Σ (β, α), 0 ≤α < 1, then |a2| ≤ p 2(1 −α) cos β (3.7) and |a3| ≤2(1 −α)cosβ, (3.8) where β ∈(−π/2, π/2).
Corollary 3.3
Corollary 3.3] and (3.8) is improvement of |a3| obtained in [10, Corollary 3.3].
Corollary 3.3] and (3.8) is improvement of |a3| obtained in [10, Corollary 3.3].
Corollary 3.7.
Corollary 3.7. Let f(z) given by (1.1) be in the class N P1,λ Σ (β, α), 0 ≤α < 1 and λ ≥1, then |a2| ≤ r 2(1 −α) cos β 2λ + 1 (3.9) and…
Corollary 3.7. Let f(z) given by (1.1) be in the class N P1,λ Σ (β, α), 0 ≤α < 1 and λ ≥1, then |a2| ≤ r 2(1 −α) cos β 2λ + 1 (3.9) and |a3| ≤2(1 −α)cosβ 2λ + 1 , (3.10) where β ∈(−π/2, π/2).
Corollary 3.9.
Corollary 3.9. Let f(z) given by (1.1) be in the class N P1,1 Σ (β, α), 0 ≤α < 1, then |a2| ≤ r 2(1 −α) cos β 3 (3.11) and |a3| ≤2(1…
Corollary 3.9. Let f(z) given by (1.1) be in the class N P1,1 Σ (β, α), 0 ≤α < 1, then |a2| ≤ r 2(1 −α) cos β 3 (3.11) and |a3| ≤2(1 −α)cosβ 3 , (3.12)
Function classes studied:
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