Convolution theorems for starlike and

Results & Lemmas (13)

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Theorem 1.1. Theorem 1.1. We have the following inclusions: (i) M (1) 1,λ ⊂S∗for λ = 1/2(log 4 −1), (ii) M (1) 1,λ ⊂K for λ = 1/log 4, (iii) M (2) 1,λ…

Theorem 1.1. We have the following inclusions: (i) M (1) 1,λ ⊂S∗for λ = 1/2(log 4 −1), (ii) M (1) 1,λ ⊂K for λ = 1/log 4, (iii) M (2) 1,λ ⊂S∗for λ = 1/[2 −4 log 2 + π2/6], (iv) M (2) 1,λ ⊂K for λ = 1/(log 4)2, (v) M1,λ ∗M1,λ ⊂S∗for 2λ2 = 1/[2 −4 log 2 + π2/6], (vi) M1,λ ∗M1,λ ⊂K for 2λ2 = 1/(log 4)2, (vii) M1,λ ∗P0 0 ⊂S∗for 2λ = 1/2(log 4 −1), (viii) M1,λ ∗P0 0 ⊂K for 2λ = 1/2 log 4.

Lemma 1.2 Lemma 1.2 ([2, Theorem 2.1]). Let α: [0, 1] →R be nonnegative, inte- grable with 1 0 α(t) dt = 1 and suppose that Λ(t) = 1 t α(s)s−2 ds…

Lemma 1.2 ([2, Theorem 2.1]). Let α : [0, 1] →R be nonnegative, inte- grable with 1 0 α(t) dt = 1 and suppose that Λ(t) = 1 t α(s)s−2 ds satisﬁes the following conditions: (i) Λ is not integrable on [0, 1], tΛ(t) is integrable on [0, 1], and positive on (0, 1). (ii) For 0 ≤γ ≤1/2, Λ(t) (1 + t)(1 −t)1+2γ is decreasing on (0, 1). For β > 0, deﬁne F(z) = z

Lemma 1.5. Lemma 1.5. Let α: [0, 1] →R be nonnegative with 1 0 α(t) dt = 1 and suppose that Λ(t) = α(t)/t satisﬁes the following conditions: (i) Λ…

Lemma 1.5. Let α : [0, 1] →R be nonnegative with 1 0 α(t) dt = 1 and suppose that Λ(t) = α(t)/t satisﬁes the following conditions: (i) Λ is not integrable on [0, 1], tΛ(t) is integrable on [0, 1], and positive on (0, 1). (ii) For 0 ≤γ ≤1/2, Λ(t) (1 + t)(1 −t)1+2γ is decreasing on (0, 1). For β > 0, let F be deﬁned by (1.3). Then F ∈Kγ for all γ ∈[0, 1/2] and for β given by β 1

Lemma 1.6 Lemma 1.6 ([2, Theorem 2.10]). If β > 0 and F ∈M1,β then F ∈Sγ (0 ≤γ ≤1/2) whenever 0 ≤β ≤ 1 −2γ 2γ + log 4.

Lemma 1.6 ([2, Theorem 2.10]). If β > 0 and F ∈M1,β then F ∈Sγ (0 ≤γ ≤1/2) whenever 0 ≤β ≤ 1 −2γ 2γ + log 4.

Lemma 1.7 Lemma 1.7 ([7]). If f, g ∈H and F, G ∈K are such that f ≺F and g ≺G, then f ∗g ≺F ∗G. Here K denotes the family of convex functions (not…

Lemma 1.7 ([7]). If f, g ∈H and F, G ∈K are such that f ≺F and g ≺G, then f ∗g ≺F ∗G. Here K denotes the family of convex functions (not necessarily normalized) in ∆. 2. Starlikeness condition for functions in M (2) 1,λ, M (1) 1,λ and M (3) 1,λ

Theorem 2.1. Theorem 2.1. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −γ 2 −2 log 4 + π2/6 + (4 log 4 −6)γ. (2.2) Then M (2) 1,λ ⊂Sγ.

Theorem 2.1. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −γ 2 −2 log 4 + π2/6 + (4 log 4 −6)γ . (2.2) Then M (2) 1,λ ⊂Sγ.

Theorem 2.7. Theorem 2.7. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −γ 2(log 4 −1) + γ(4 −4 log 2 −π2/6). (2.8) Then M (1) 1,λ ⊂Sγ.

Theorem 2.7. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −γ 2(log 4 −1) + γ(4 −4 log 2 −π2/6). (2.8) Then M (1) 1,λ ⊂Sγ.

Theorem 2.10. Theorem 2.10. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −γ 2(4 log 2 −1 −π2/6) −γ(10 + 3ζ(3) −16 log 2 −π2/3), (2.11) where ζ(3) =…

Theorem 2.10. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −γ 2(4 log 2 −1 −π2/6) −γ(10 + 3ζ(3) −16 log 2 −π2/3), (2.11) where ζ(3) = 1.202056903 . . . denotes the Riemann zeta value at 3. Then M (3) 1,λ ⊂Sγ.

Theorem 3.1. Theorem 3.1. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −2γ (log 4)2 + 2γ log 4. (3.2) Then M (2) 1,λ ⊂Kγ.

Theorem 3.1. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −2γ (log 4)2 + 2γ log 4. (3.2) Then M (2) 1,λ ⊂Kγ.

Theorem 3.4. Theorem 3.4. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −2γ log 4 + 2γ. Then M (1) 1,λ ⊂Kγ.

Theorem 3.4. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −2γ log 4 + 2γ . Then M (1) 1,λ ⊂Kγ.

Theorem 3.5. Theorem 3.5. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −γ 2(3 −log 16) + γ(2 −2 log 16 + π2/3). (3.6) Then M (3) 1,λ ⊂Kγ.

Theorem 3.5. Let γ ∈[0, 1/2] and let λ > 0 be given by λ = 1 −γ 2(3 −log 16) + γ(2 −2 log 16 + π2/3). (3.6) Then M (3) 1,λ ⊂Kγ.

Theorem 4.7. Theorem 4.7. Let γ ∈[0, 1/2] and β < 1. Then: (i) M1,α ∗P0 β ⊂Sγ whenever 2α(1 −β) = 1 −γ 2(log 4 −1) + γ(4 −4 log 2 −π2/6). (ii) M1,α ∗P0…

Theorem 4.7. Let γ ∈[0, 1/2] and β < 1. Then: (i) M1,α ∗P0 β ⊂Sγ whenever 2α(1 −β) = 1 −γ 2(log 4 −1) + γ(4 −4 log 2 −π2/6). (ii) M1,α ∗P0 β ⊂Kγ whenever 2α(1 −β) = 1 −2γ 2γ + log 4.

Theorem 4.8. Theorem 4.8. Let γ ∈[0, 1/2] and α, β > 0. Then: (i) M1,α ∗M1,β ⊂Sγ whenever 2αβ = 1 −γ 2 −2 log 4 + π2/6 + (4 log 4 −6)γ. (ii) M1,α ∗M1,β…

Theorem 4.8. Let γ ∈[0, 1/2] and α, β > 0. Then: (i) M1,α ∗M1,β ⊂Sγ whenever 2αβ = 1 −γ 2 −2 log 4 + π2/6 + (4 log 4 −6)γ . (ii) M1,α ∗M1,β ⊂Kγ whenever 2αβ = 1 −2γ (log 4)2 + 2γ log 4.