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Abstract
In this paper, we introduce two general subclasses of analytic functions by means of the principle of subordination and investigate the coefficient bounds for functions in theese classes. The well-known results are obtained as a corollary of our main results. Especially, we improve the results of Altintas and Kilic.
Results & Lemmas (16)
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Theorem 1.
Theorem 1. If f ∈QCV (λ, A, B), then |an| ≤ 1 1 + (n −1) λ 1 + (n −1) (A −B) 1 −B (n = 2, 3,...).
Theorem 1. If f ∈QCV (λ, A, B) , then |an| ≤ 1 1 + (n −1) λ 1 + (n −1) (A −B) 1 −B (n = 2, 3, . . .) .
Theorem 2.
Theorem 2. If f ∈QST (λ, A, B), then |an| ≤ n 1 + (n −1) λ 1 + (n −1) (A −B) 1 −B (n = 2, 3,...). In this work, we obtain coefficient…
Theorem 2. If f ∈QST (λ, A, B) , then |an| ≤ n 1 + (n −1) λ 1 + (n −1) (A −B) 1 −B (n = 2, 3, . . .) . In this work, we obtain coefficient bounds for functions in the comprehensive subclasses Kλ,δ (ϕ, ψ) and Sλ,δ (ϕ, ψ) of analytic functions. Our results improve the results of Altınta¸s and Kılı¸c [1] (Theorem 1 and Theorem 2). 2. Main results
Lemma 1.
Lemma 1. [9] Let the function Φ given by Φ (z) = ∞ X n=1 Anzn (z ∈D) be convex in D. Also let the function Ψ given by Ψ(z) = ∞ X n=1 Bnzn…
Lemma 1. [9] Let the function Φ given by Φ (z) = ∞ X n=1 Anzn (z ∈D) be convex in D. Also let the function Ψ given by Ψ(z) = ∞ X n=1 Bnzn (z ∈D) be holomorphic in D. If
Lemma 2.
Lemma 2. [10] Let f ∈K (ψ) and be of the form (1.1), then |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) n! (n = 2, 3,...).
Lemma 2. [10] Let f ∈K (ψ) and be of the form (1.1), then |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) n! (n = 2, 3, . . .) .
Lemma 3.
Lemma 3. [10] Let f ∈S∗(ψ) and be of the form (1.1), then |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) (n −1)! (n = 2, 3,...).
Lemma 3. [10] Let f ∈S∗(ψ) and be of the form (1.1), then |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) (n −1)! (n = 2, 3, . . .) .
Theorem 3.
Theorem 3. Let f ∈Kλ,δ (ϕ, ψ) and be of the form (1.1), then [1 + (n −1) (λ −δ + 2λδ) + (n −1) (n −2) λδ] |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) n!…
Theorem 3. Let f ∈Kλ,δ (ϕ, ψ) and be of the form (1.1), then [1 + (n −1) (λ −δ + 2λδ) + (n −1) (n −2) λδ] |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) n! + |ϕ′(0)| n 1 +
Theorem 4.
Theorem 4. Let f ∈Sλ,δ (ϕ, ψ) and be of the form (1.1), then [1 + (n −1) (λ −δ + 2λδ) + (n −1) (n −2) λδ] |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) (n…
Theorem 4. Let f ∈Sλ,δ (ϕ, ψ) and be of the form (1.1), then [1 + (n −1) (λ −δ + 2λδ) + (n −1) (n −2) λδ] |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) (n −1)! + |ϕ′(0)| 1 + n−2
Corollary 1.
Corollary 1. Let f ∈QK (ϕ, ψ) and be of the form (1.1), then |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) n2 (n −1)! + |ϕ′(0)| n2 1 + n−2
Corollary 1. Let f ∈QK (ϕ, ψ) and be of the form (1.1), then |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) n2 (n −1)! + |ϕ′(0)| n2 1 + n−2
Corollary 2.
Corollary 2. Let f ∈C (ϕ, ψ) and be of the form (1.1), then |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) n! + |ϕ′(0)| n 1 + n−2
Corollary 2. Let f ∈C (ϕ, ψ) and be of the form (1.1), then |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) n! + |ϕ′(0)| n 1 + n−2
Corollary 3.
Corollary 3. Let f ∈CS (ϕ, ψ) and be of the form (1.1), then |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) (n −1)! + |ϕ′(0)| 1 + n−2 X
Corollary 3. Let f ∈CS (ϕ, ψ) and be of the form (1.1), then |an| ≤ n−2 Q j=0 (j + |ψ′(0)|) (n −1)! + |ϕ′(0)| 1 + n−2 X
Corollary 4.
Corollary 4. [4] Let f ∈C(α, β) (0 ≤α, β < 1) and be of the form (1.1), then |an| ≤2 (3 −2β) (4 −2β) · · · (n −2β) n! [n (1 −α) + (α −β)]…
Corollary 4. [4] Let f ∈C(α, β) (0 ≤α, β < 1) and be of the form (1.1), then |an| ≤2 (3 −2β) (4 −2β) · · · (n −2β) n! [n (1 −α) + (α −β)] (n = 2, 3, . . .) . Letting δ = 0, ϕ (z) = 1 + Az 1 + Bz (−1 ≤B < A ≤1) , ψ (z) = 1 + z 1 −z in Theorem 3 and Theorem 4, we obtain the following consequences, respectively.
Corollary 5.
Corollary 5. Let f ∈QCV (λ, A, B) and be of the form (1.1), then |an| ≤ 1 1 + (n −1) λ 1 + (n −1) (A −B) 2 (n = 2, 3,...).
Corollary 5. Let f ∈QCV (λ, A, B) and be of the form (1.1), then |an| ≤ 1 1 + (n −1) λ 1 + (n −1) (A −B) 2 (n = 2, 3, . . .) .
Corollary 6.
Corollary 6. Let f ∈QST (λ, A, B) and be of the form (1.1), then |an| ≤ n 1 + (n −1) λ 1 + (n −1) (A −B) 2 (n = 2, 3,...).
Corollary 6. Let f ∈QST (λ, A, B) and be of the form (1.1), then |an| ≤ n 1 + (n −1) λ 1 + (n −1) (A −B) 2 (n = 2, 3, . . .) .
Corollary 7.
Corollary 7. [8] Let f ∈C and be of the form (1.1), then |an| ≤n (n = 2, 3,...).
Corollary 7. [8] Let f ∈C and be of the form (1.1), then |an| ≤n (n = 2, 3, . . .) .
Corollary 8.
Corollary 8. [8] Let f ∈CS and be of the form (1.1), then |an| ≤n2 (n = 2, 3,...). Letting λ = 1, A = 1, B = −1 in Corollary 5, we have…
Corollary 8. [8] Let f ∈CS and be of the form (1.1), then |an| ≤n2 (n = 2, 3, . . .) . Letting λ = 1, A = 1, B = −1 in Corollary 5, we have following consequence.
Corollary 9. · radius
Corollary 9. [6] Let f ∈Q and be of the form (1.1), then |an| ≤1 (n = 2, 3,...). References [1] Altınta¸s O, Kılı¸c ¨O ¨O. Coefficient…
Corollary 9. [6] Let f ∈Q and be of the form (1.1), then |an| ≤1 (n = 2, 3, . . .) . References [1] Altınta¸s O, Kılı¸c ¨O ¨O. Coefficient estimates for a class containing quasi-convex functions. Turkish Journal of Mathematics 2018; 42 (5): 2819–2825. doi:10.3906/mat-1805-90 [2] Janowski W. Some extremal problems for certain families of analytic functions I. Annales Polonici Math- ematici 1973; 28: 297–326. [3] Kim YC, Choi JH, Sugawa T. Coefficient bounds and convolution properties for certain clas
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