Abstract
In this article, we define and investigate new families of certain subclasses of
meromorphic functions of complex order. Considering the new subclasses, several
properties for certain integral operators are derived.
2010 Mathematics Subject Classification: 30C45.
Results & Lemmas (15)
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Corollary 1.1
Corollary 1.1 If f Î Σ satisfies the following inequality zf ′′(z) f ′(z) −2zf ′(z) f(z) < (1 −δ) (3 −δ) 2 −δ, 0 ≤δ < 1, (1:17)…
Corollary 1.1 If f Î Σ satisfies the following inequality zf ′′(z) f ′(z) −2zf ′(z) f(z) < (1 −δ) (3 −δ) 2 −δ , 0 ≤δ < 1, (1:17) then f Î Σ*(δ).
Corollary 1.2
Corollary 1.2 If f Î Σ satisfies the following inequality zf ′′(z) f ′(z) −zf ′(z) f(z) + 1 < 1 2, (1:18) then f Î Σ*.
Corollary 1.2 If f Î Σ satisfies the following inequality zf ′′(z) f ′(z) −zf ′(z) f(z) + 1 < 1 2, (1:18) then f Î Σ*.
Corollary 1.3
Corollary 1.3 If f Î Σ satisfies the following inequality ℜ zf ′(z) f(z) 2zf ′(z) f(z) −zf ′′(z) f ′(z) −1 > −1 2, (1:19) then f Î Σ*.…
Corollary 1.3 If f Î Σ satisfies the following inequality ℜ zf ′(z) f(z) 2zf ′(z) f(z) −zf ′′(z) f ′(z) −1 > −1 2, (1:19) then f Î Σ*. In this article, we derive several properties for the integral operators Hn(z) and Hγ1,...,γn(z) of the subclasses given by (1.5) and Definitions 1.1 to 1.6.
Theorem 2.1
Theorem 2.1 For i Î 1,..., n, let gi > 0, fi Î Σ and zf ′′ i (z) f ′ i (z) −2zf ′ i (z) fi(z) < (1 −δ) (3 −δ) 2 −δ, (0 ≤δ < 1).
Theorem 2.1 For i Î {1,..., n}, let gi > 0, fi Î Σ and zf ′′ i (z) f ′ i (z) −2zf ′ i (z) fi(z) < (1 −δ) (3 −δ) 2 −δ , (0 ≤δ < 1) .
Corollary 2.2
Corollary 2.2 For i Î 1,..., n, let gi > 0, fi Î Σ and zf ′′ i (z) f ′ i (z) −2zf ′ i (z) fi(z) < 3 2. (2:13) If
Corollary 2.2 For i Î {1,..., n}, let gi > 0, fi Î Σ and zf ′′ i (z) f ′ i (z) −2zf ′ i (z) fi(z) < 3 2. (2:13) If
Theorem 2.3
Theorem 2.3 For i Î 1,..., n, let gi > 0, fi Î Σ and zf ′′ i (z) f ′ i (z) −zf ′ i (z) fi(z) + 1 < 1 2. (2:15) If
Theorem 2.3 For i Î {1,..., n}, let gi > 0, fi Î Σ and zf ′′ i (z) f ′ i (z) −zf ′ i (z) fi(z) + 1 < 1 2. (2:15) If
Theorem 2.4
Theorem 2.4 For i Î 1,..., n, let gi > 0, fi Î Σ and ℜ
Theorem 2.4 For i Î {1,..., n}, let gi > 0, fi Î Σ and ℜ
Theorem 2.5
Theorem 2.5 For i Î 1,..., n, let gi > 0 and fi ∈⋆ b(δi) (0 ≤δ < 1 and b ∈C 0 ). If 0 < n i=1 γi (1 −δi) ≤1, (2:19) then Hn(z)is in the…
Theorem 2.5 For i Î {1,..., n}, let gi > 0 and fi ∈⋆ b(δi) (0 ≤δ < 1 and b ∈C\{0}). If 0 < n i=1 γi (1 −δi) ≤1, (2:19) then Hn(z)is in the class F1(μ, b), μ = 1 − n i=1 γi (1 −δi).
Theorem 2.6
Theorem 2.6 For i Î 1,..., n, let gi > 0 and fi ∈⋆U(α, δ, b) (a ≥0, δ Î [-1,1), a + δ ≥0 and b ∈C 0 ). If n i=1 γi ≤1, (2:25) then…
Theorem 2.6 For i Î {1,..., n}, let gi > 0 and fi ∈⋆U(α, δ, b) (a ≥0, δ Î [-1,1), a + δ ≥0 and b ∈C\{0}). If n i=1 γi ≤1, (2:25) then Hn(z)is in the class F2(α, δ, b).
Theorem 2.7
Theorem 2.7 For i Î 1,..., n, let gi > 0 and fi ∈⋆UH(α, b) (a > 0 and b ∈C 0 ). If Mohammed and Darus Journal of Inequalities and…
Theorem 2.7 For i Î {1,..., n}, let gi > 0 and fi ∈⋆UH(α, b) (a > 0 and b ∈C\{0}). If Mohammed and Darus Journal of Inequalities and Applications 2011, 2011:121 http://www.journalofinequalitiesandapplications.com/content/2011/1/121 Page 7 of 12
Theorem 3.1
Theorem 3.1 For i Î 1,..., n, let gi > 0, fi Î Σ and n i=1 γi > 2 1 −δ, (0 ≤δ < 1). (3:1) If fi ∈k(δ), then Hγ1,...,γn(z) ∈N(β), b > 1.…
Theorem 3.1 For i Î {1,..., n}, let gi > 0, fi Î Σ and n i=1 γi > 2 1 −δ , (0 ≤δ < 1) . (3:1) If fi ∈k(δ), then Hγ1,...,γn(z) ∈N(β), b > 1. Mohammed and Darus Journal of Inequalities and Applications 2011, 2011:121 http://www.journalofinequalitiesandapplications.com/content/2011/1/121 Page 8 of 12
Theorem 3.2
Theorem 3.2 For i Î 1,..., n, let gi > 0, fi Î Σ and 1 < n i=1 γi < 2. (3:12) If Hγ1,...,γn(z) ∈⋆(δ), then Hγ1,...,γn(z) ∈N(β), b > 1.
Theorem 3.2 For i Î {1,..., n}, let gi > 0, fi Î Σ and 1 < n i=1 γi < 2. (3:12) If Hγ1,...,γn(z) ∈⋆(δ), then Hγ1,...,γn(z) ∈N(β), b > 1.
Theorem 3.3
Theorem 3.3 For i Î 1,..., n, let gi > 0 fi Î ΣKb(δi) (0 ≤δ < 1 and b ∈C 0 )). If 0 < n i=1 γi (1 −δi) ≤1, (3:18) then Hγ1,...,γn(z)is in…
Theorem 3.3 For i Î {1,..., n}, let gi > 0 fi Î ΣKb(δi) (0 ≤δ < 1 and b ∈C\{0})). If 0 < n i=1 γi (1 −δi) ≤1, (3:18) then Hγ1,...,γn(z)is in the class F1(μ, b), μ = 1 − n i=1 γi (1 −δi).
Theorem 3.4
Theorem 3.4 For i Î 1,..., n, let gi > 0 and fi ∈KU(α, δ, b) (a ≥0, δ Î [-1,1), a + δ ≥0 and b ∈C 0 )). If n i=1 γi ≤1, (3:19) then…
Theorem 3.4 For i Î {1,..., n}, let gi > 0 and fi ∈KU(α, δ, b) (a ≥0, δ Î [-1,1), a + δ ≥0 and b ∈C\{0})). If n i=1 γi ≤1, (3:19) then Hγ1,...,γn(z)is in the class F2(α, δ, b).
Theorem 3.5
Theorem 3.5 For i Î 1,..., n, let gi > 0 and fi ∈KUH(α, b) (a ≥0, and b ∈C 0 )). If n i=1 γi ≤1, (3:20) then Hγ1,...,γn(z)is in the…
Theorem 3.5 For i Î {1,..., n}, let gi > 0 and fi ∈KUH(α, b) (a ≥0, and b ∈C\{0})). If n i=1 γi ≤1, (3:20) then Hγ1,...,γn(z)is in the class F3(α, δ, b). Acknowledgements The above study was supported by MOHE grant: UKM-ST-06-FRGS0244-2010. Authors’ contributions AM is currently a PhD student under supervision of MD and jointly worked on deriving the results. All the authors read and approved the final manuscript. Mohammed and Darus Journal of Inequalities and Applications 2011, 2011:121 htt
Definitions (9)
Def 1.1
Definition 1.1 Let a function f Î Σ be analytic in U∗. Then f is in the class ⋆ b(δ) if, and only if, f satisfies ℜ 1 −1 b zf ′(z) f(z)…
Definition 1.1 Let a function f Î Σ be analytic in U∗. Then f is in the class ⋆ b(δ) if, and only if, f satisfies ℜ 1 −1 b zf ′(z) f(z) + 1
Def 1.2
Definition 1.2 Let a function f Î Σ be analytic in U∗. Then f is in the class ΣKb(δ) if, and only if, f satisfies ℜ 1 −1 b zf ′′(z) f…
Definition 1.2 Let a function f Î Σ be analytic in U∗. Then f is in the class ΣKb(δ) if, and only if, f satisfies ℜ 1 −1 b zf ′′(z) f ′(z) + 2 > δ,
Def 1.3
Definition 1.3 Let a function f Î Σ be analytic in U∗. Then f is in the class ⋆U(α, δ, b) if, and only if, f satisfies ℜ 1 −1 b zf ′(z)…
Definition 1.3 Let a function f Î Σ be analytic in U∗. Then f is in the class ⋆U(α, δ, b) if, and only if, f satisfies ℜ 1 −1 b zf ′(z) f(z) + 1 > α
Def 1.4
Definition 1.4 Let a function f Î Σ be analytic in U∗. Then f is in the class KU(α, δ, b) if, and only if, f satisfies ℜ 1 −1 b zf…
Definition 1.4 Let a function f Î Σ be analytic in U∗. Then f is in the class KU(α, δ, b) if, and only if, f satisfies ℜ 1 −1 b zf ′′(z) f ′(z) + 2 > α
Def 1.5
Definition 1.5 Let a function f Î Σ be analytic in U∗. Then f is in the class ⋆UH(α, b) if, and only if, f satisfies 1 −1 b zf ′(z)…
Definition 1.5 Let a function f Î Σ be analytic in U∗. Then f is in the class ⋆UH(α, b) if, and only if, f satisfies 1 −1 b zf ′(z) f(z) + 1 −2α( √ 2 −1)
Def 1.6
Definition 1.6 Let a function f Î Σ be analytic in U∗. Then f is in the class KUH(α, b) if, and only if, f satisfies 1 −1 b zf ′′(z)…
Definition 1.6 Let a function f Î Σ be analytic in U∗. Then f is in the class KUH(α, b) if, and only if, f satisfies 1 −1 b zf ′′(z) f ′(z) + 2 −2α( √ 2 −1)
Def 1.7
Definition 1.7 Let a function f Î Σ be analytic in U∗. Then f is in the class F1(δ, b), if, and only if, f satisfies ℜ
Definition 1.7 Let a function f Î Σ be analytic in U∗. Then f is in the class F1(δ, b), if, and only if, f satisfies ℜ
Def 1.8
Definition 1.8 Let a function f Î Σ be analytic in U∗. Then f is in the class F2(α, δ, b) if, and only if, f satisfies ℜ
Definition 1.8 Let a function f Î Σ be analytic in U∗. Then f is in the class F2(α, δ, b) if, and only if, f satisfies ℜ
Def 1.9
Definition 1.9 Let a function f Î Σ be analytic in U∗. Then f is in the class F3(α, b) if, and only if, f satisfies 1 −1 b
Definition 1.9 Let a function f Î Σ be analytic in U∗. Then f is in the class F3(α, b) if, and only if, f satisfies 1 −1 b
Function classes studied:
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