Results & Lemmas (11)
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Theorem 4.
Theorem 4. Let v given by (1) be in the class N Ξ,δ(ς, n; x). Then |am+1| ≤|P(x)| p |P(x)| (| (1 + 2mδ)n(1 + 2mς) −(1 + mδ)2n(1 + mς)2…
Theorem 4. Let v given by (1) be in the class N Ξ,δ(ς, n; x). Then |am+1| ≤|P(x)| p |P(x)| (|{(1 + 2mδ)n(1 + 2mς) −(1 + mδ)2n(1 + mς)2}P2(x)−2(1 + mδ)2n(1 + mς)2Q(x)|)−1 2 (10) and |a2m+1| ≤ P2(x) (1 + mδ)2n(1 + mς)2 + |P(x)| (1 + 2mδ)n(1 + 2mς) . (11)
Corollary 5.
Corollary 5. Let v ∈N Ξ,1(ς, n; x) = N Ξ(ς, n; x). Then, |am+1| ≤|P(x)| p |P(x)| ( q | (1 + 2m)n(1 + 2mς) −(1 + m)2n(1 + mς)2 P2(x)−(1 +…
Corollary 5. Let v ∈N Ξ,1(ς, n; x) = N Ξ(ς, n; x). Then, |am+1| ≤|P(x)| p |P(x)| ( q |{(1 + 2m)n(1 + 2mς) −(1 + m)2n(1 + mς)2}P2(x)−(1 + m)2n+1(1 + mς)2Q(x)|)−1 2 (27) and |a2m+1| ≤ P2(x) (1 + m)2n(1 + mς)2 + |P(x)|
Corollary 6.
Corollary 6. Let v ∈N Ξ,1(ς, 0; x) = N Ξ(ς; x). Then, |am+1| ≤ |P(x)| p |P(x)| p |m2ς2P2(x) + 2(1 + mς)2Q(x)| (29) and |a2m+1| ≤ P2(x) (1 +…
Corollary 6. Let v ∈N Ξ,1(ς, 0; x) = N Ξ(ς; x). Then, |am+1| ≤ |P(x)| p |P(x)| p |m2ς2P2(x) + 2(1 + mς)2Q(x)| (29) and |a2m+1| ≤ P2(x) (1 + mς)2 + |P(x)| (1 + 2mς) .
Corollary 7.
Corollary 7. Let v ∈N Ξ,1(1, 0; x) = N Ξ(x). Then, |am+1| ≤ |P(x)| p |P(x)| p |m2P2(x) + 2(1 + m)2Q(x)| (31) and |a2m+1| ≤ P2(x) (1 + m)2 +…
Corollary 7. Let v ∈N Ξ,1(1, 0; x) = N Ξ(x). Then, |am+1| ≤ |P(x)| p |P(x)| p |m2P2(x) + 2(1 + m)2Q(x)| (31) and |a2m+1| ≤ P2(x) (1 + m)2 + |P(x)| 1 + 2m . (32)
Theorem 8.
Theorem 8. Let v given by (1) belongs to the class N Ξ,δ(ς, n; x). Then, |a2m+1 −ηa2 m+1| ≤ ( |P(x)| (1+2mς)(1+2mδ)n, 0 ≤|t(λ; x)| < A, (1…
Theorem 8. Let v given by (1) belongs to the class N Ξ,δ(ς, n; x). Then, |a2m+1 −ηa2 m+1| ≤ ( |P(x)| (1+2mς)(1+2mδ)n , 0 ≤|t(λ; x)| < A, (1 + m)|P(x)||t(λ; x)|, |t(λ; x)| ≥A, (33) where A = 1 (1 + m)(1 + 2mς)(1 + 2mδ)n ,
Corollary 9.
Corollary 9. Let v ∈N Ξ,1(ς, n; x) = N Ξ(ς, n; x) and λ ∈R. Then, |a2m+1 −ηa2 m+1| ≤ |P(x)| (1 + 2mς)(1 + 2m)n, 0 ≤|t(λ; x)| < B, (1…
Corollary 9. Let v ∈N Ξ,1(ς, n; x) = N Ξ(ς, n; x) and λ ∈R. Then, |a2m+1 −ηa2 m+1| ≤ |P(x)| (1 + 2mς)(1 + 2m)n , 0 ≤|t(λ; x)| < B, (1 + m)|P(x)||t(λ; x)|, |t(λ; x)| ≥B, (34)
Corollary 10.
Corollary 10. Let v ∈N Ξ,1(ς, 0; x) = N Ξ(ς; x) and λ ∈R. Then, |a2m+1 −ηa2 m+1| ≤ |P(x)| (1 + 2mς), 0 ≤|t(λ; x)| < C, (1 +…
Corollary 10. Let v ∈N Ξ,1(ς, 0; x) = N Ξ(ς; x) and λ ∈R. Then, |a2m+1 −ηa2 m+1| ≤ |P(x)| (1 + 2mς), 0 ≤|t(λ; x)| < C, (1 + m)|P(x)||t(λ; x)|, |t(λ; x)| ≥C, (35) where C =
Corollary 11.
Corollary 11. Let v ∈N Ξ,1(1, 0; x) = N Ξ(x) and λ ∈R. Then, |a2m+1 −ηa2 m+1| ≤ |P(x)| 1 + 2m, 0 ≤|t(λ; x)| < D, (1 + m)|P(x)||t(λ;…
Corollary 11. Let v ∈N Ξ,1(1, 0; x) = N Ξ(x) and λ ∈R. Then, |a2m+1 −ηa2 m+1| ≤ |P(x)| 1 + 2m, 0 ≤|t(λ; x)| < D, (1 + m)|P(x)||t(λ; x)|, |t(λ; x)| ≥D, (36) where D =
Corollary 12.
Corollary 12. Let v ∈N Ξ,1(ς, n; x) = N Ξ(ς, n; x). Then, |a2m+1 −ηa2 m+1| ≤ |P(x)| (1 + 2mς)(1 + 2m)n. (37)
Corollary 12. Let v ∈N Ξ,1(ς, n; x) = N Ξ(ς, n; x). Then, |a2m+1 −ηa2 m+1| ≤ |P(x)| (1 + 2mς)(1 + 2m)n . (37)
Corollary 13.
Corollary 13. Let v ∈N Ξ,1(ς, 0; x) = N Ξ(ς; x). Then, |a2m+1 −ηa2 m+1| ≤ |P(x)| (1 + 2mς). (38)
Corollary 13. Let v ∈N Ξ,1(ς, 0; x) = N Ξ(ς; x). Then, |a2m+1 −ηa2 m+1| ≤ |P(x)| (1 + 2mς) . (38)
Corollary 14.
Corollary 14. Let v ∈N Ξ,δ(1, 0; x) = N Ξ,δ(x). Then, |a2m+1 −ηa2 m+1| ≤|P(x)| 1 + 2m. (39) References [1] S¸. Altinkaya, S. Yal¸cin,…
Corollary 14. Let v ∈N Ξ,δ(1, 0; x) = N Ξ,δ(x). Then, |a2m+1 −ηa2 m+1| ≤|P(x)| 1 + 2m . (39) References [1] S¸. Altinkaya, S. Yal¸cin, Initial coefficient bounds for a general class of bi-univalent functions, Int. J. Anal., (2014), Article ID 867871, 4 pages. [2] S¸. Altinkaya, S. Yal¸cin, Coefficient estimates for two new subclasses of bi-univalent functions with respect to symmetric points, J. Func. Spaces, (2015), Article ID 145242, 5 pages. [3] D.A. Brannan, J. Clunie (Eds), Aspects of Contempor
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