Abstract
In this paper, some geometric properties of normalized Mittag-Leffler functions are
investigated. We focus on starlikeness of order 2μ + η – 1 and convexity in the
direction of imaginary axis. In addition, we study pre-starlikeness of Mittag-Leffler
functions. The results are obtained by using the positivity technique.
MSC: 33E12; 30C45
Results & Lemmas (14)
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Lemma 2.1
Lemma 2.1 ([18]) Let ak ∞ k=1 be a sequence of positive numbers such that a1 = 1. If, for 0 ≤μ < 1, (1) (1 – μ)a1 ≥(2 – μ)a2 ≥2(μ+1)(3 –…
Lemma 2.1 ([18]) Let {ak}∞ k=1 be a sequence of positive numbers such that a1 = 1. If, for 0 ≤μ < 1, (1) (1 – μ)a1 ≥(2 – μ)a2 ≥2(μ+1)(3 – μ)a3, (2) (k – 1 – μ)(k – μ)ak ≥k(k + 1 – μ)ak+1, ∀k ≥3. Then f (z) = z + ∞ k=2 akzk ∈S∗(μ).
Lemma 2.2
Lemma 2.2 ([20]) Let η ≥0, μ ∈R such that 0 < μ + η < 1 and n ∈N. If d0 = d1 = 1 and d2k = d2k+1 = (1+η)n–kn! (n–k)!(1+η)n. (μ+η)k k! for 1…
Lemma 2.2 ([20]) Let η ≥0, μ ∈R such that 0 < μ + η < 1 and n ∈N. If d0 = d1 = 1 and d2k = d2k+1 = (1+η)n–kn! (n–k)!(1+η)n . (μ+η)k k! for 1 ≤k ≤n, then (i) n k=0 dk cos(kθ) > 0 ⇔μ + η ≤μ∗( 1 2) = 0.691556... , (ii) 2n+1 k=1 sin(kθ) > 0 ⇔μ + η ≤μ∗( 1 2), (iii) 2n k=1 sin(kθ) > 0 for μ + η ≤1+η 2 ,
Lemma 2.3
Lemma 2.3 ([31]) Let 0 ≤η ≤2μ∗ 0 – 1, μ ∈R such that 0 < μ + η < 1 and n ∈N. If ak ∞ k=1 is a decreasing sequence of non-negative numbers…
Lemma 2.3 ([31]) Let 0 ≤η ≤2μ∗ 0 – 1, μ ∈R such that 0 < μ + η < 1 and n ∈N. If {ak}∞ k=1 is a decreasing sequence of non-negative numbers satisfying a0 > 0 and k(n – k + 1 + η)a2k ≤(n – k + 1)(k + μ + η – 1)a2k–1 for 1 ≤k ≤n, then, for all 0 < θ < π, n k=0 ak sinkθ > 0 ⇔ μ + η ≤1 + η 2 .
Lemma 2.4
Lemma 2.4 ([31]) Let 0 ≤η ≤2μ∗ 0 – 1 and –η < μ ≤1–η 2, a1 = 1, ak ≥0 satisfy k(1 + η)(1 – μ – η) – 1 + 2μ + η
Lemma 2.4 ([31]) Let 0 ≤η ≤2μ∗ 0 – 1 and –η < μ ≤1–η 2 , a1 = 1, ak ≥0 satisfy k(1 + η)(1 – μ – η) – 1 + 2μ + η
Lemma 2.5
Lemma 2.5 ([31]) Let μ ∈R and η ≥0 such that 0 < μ + η < 1, and let a1 = 1 and ak ≥0 satisfy 0 ≤nan ≤··· ≤(k + 1)ak+1 ≤kak ≤··· ≤3a3 ≤2a2…
Lemma 2.5 ([31]) Let μ ∈R and η ≥0 such that 0 < μ + η < 1, and let a1 = 1 and ak ≥0 satisfy 0 ≤nan ≤··· ≤(k + 1)ak+1 ≤kak ≤··· ≤3a3 ≤2a2 ≤μ + η μ∗ 0 , μ + η ∈ 0,μ∗ 0
Theorem 3.1
Theorem 3.1 Let α ≥1, β ≥1. Then Eα,β(z) ∈R[ρ,μ] for 0 ≤μ < 1 and Γ (α + β) Γ (β) ≥ ⎧ ⎪⎪⎨ ⎪⎪⎩ T1(ρ,μ), 0 ≤ρ ≤ρ1(μ), max T1(ρ,μ),T2(ρ,μ) 2Γ…
Theorem 3.1 Let α ≥1, β ≥1. Then Eα,β(z) ∈R[ρ,μ] for 0 ≤μ < 1 and Γ (α + β) Γ (β) ≥ ⎧ ⎪⎪⎨ ⎪⎪⎩ T1(ρ,μ), 0 ≤ρ ≤ρ1(μ), max{T1(ρ,μ),T2(ρ,μ) 2Γ 2(α+β) Γ (β)Γ (2α+β),T3(ρ,μ) Γ (α(k–1)+β) Γ (αk+β) Γ (α+β) Γ (β) },
Theorem 3.2
Theorem 3.2 Let 0 ≤μ < 1, α ≥1, β ≥1. If Γ (α+β) Γ (β) ≥2(2 – μ), then Eα,β(z) is pre-starlike of order μ in U.
Theorem 3.2 Let 0 ≤μ < 1, α ≥1, β ≥1. If Γ (α+β) Γ (β) ≥2(2 – μ), then Eα,β(z) is pre-starlike of order μ in U.
Corollary 3.3
Corollary 3.3 Let α ≥1, β ≥1. Then Eα,β(z) ∈C if Γ (α+β) Γ (β) ≥4.
Corollary 3.3 Let α ≥1, β ≥1. Then Eα,β(z) ∈C if Γ (α+β) Γ (β) ≥4.
Corollary 3.6
Corollary 3.6 Let α ≥1, β ≥1. Then Eα,β(z) ∈S∗( 1 2) if Γ (α+β) Γ (β) ≥3.
Corollary 3.6 Let α ≥1, β ≥1. Then Eα,β(z) ∈S∗( 1 2) if Γ (α+β) Γ (β) ≥3.
Theorem 3.9
Theorem 3.9 Let μ ≥1, η ≥–μ+√ μ2+4μ–4 2 with α ≥1, β ≥2 If M1 = (1 + η)(1 – μ – η) > 0 and M2 = +2μ + η – 1 > 0, then Eα,β(z) is starlike…
Theorem 3.9 Let μ ≥1, η ≥–μ+√ μ2+4μ–4 2 with α ≥1, β ≥2 If M1 = (1 + η)(1 – μ – η) > 0 and M2 = +2μ + η – 1 > 0, then Eα,β(z) is starlike of order 2μ + η – 1.
Lemma 2.4.
Lemma 2.4. Using the above relation and simple computations yields k(1 + η)(1 – μ – η) – 1 + 2μ + η
Lemma 2.4. Using the above relation and simple computations yields k(1 + η)(1 – μ – η) – 1 + 2μ + η
Theorem 3.10
Theorem 3.10 Let μ ≥1, η ≥–μ+√ μ2+4μ–4 2, 2μ + η > 1, α ≥1, β ≥2, a1 = 1, ak ≥0 satisfy kak – (k + 1)ak+1 ≥0, k = 1,2,3,...,n – 1, (n – k +…
Theorem 3.10 Let μ ≥1, η ≥–μ+√ μ2+4μ–4 2 , 2μ + η > 1, α ≥1, β ≥2, a1 = 1, ak ≥0 satisfy kak – (k + 1)ak+1 ≥0, k = 1,2,3,...,n – 1, (n – k + 1)(k + μ + η – 1)(2k – 1)a2k–1 ≥2k2(n – k + 1 + η)a2k, k = 4,5,..., n + 3 2 , (3.4) for k ≥4. Then (Eα,β)n(z) = n k=4 akzk is convex in the direction of imaginary axis.
Lemma 2.3.
Lemma 2.3. Therefore, by using the minimum principle for harmonic functions under the conditions μ + η ∈(0, 1+η 2 ], Im zf ′ n(z)
Lemma 2.3. Therefore, by using the minimum principle for harmonic functions under the conditions μ + η ∈(0, 1+η 2 ], Im zf ′ n(z)
Theorem 3.11
Theorem 3.11 Let μ ∈R and η ≥0 such that 2μ + η > 1, and let a1 = 1 and ak ≥0 satisfy 0 ≤nan ≤··· ≤(k + 1)ak+1 ≤kak ≤··· ≤3a3 ≤2a2 ≤μ + η…
Theorem 3.11 Let μ ∈R and η ≥0 such that 2μ + η > 1, and let a1 = 1 and ak ≥0 satisfy 0 ≤nan ≤··· ≤(k + 1)ak+1 ≤kak ≤··· ≤3a3 ≤2a2 ≤μ + η μ∗ 0 , μ + η ∈ 0,μ∗ 0
Function classes studied:
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