Starlikeness and convexity of integral operators involving Mittag-Leffler functions

Lemma 1.1. ( [8]). Let Φ(u, v) be a complex valued function, Φ : D →C, (D ⊂C2) and let u = u1 + iu2 and v = v1 + iv2. Suppose that the function Φ(u, v) satisfies (i) Φ(u, v) is continuous in D; (ii) (1, 0) ∈D and Re(Φ(1, 0)) > 0; (iii) Re(Φ(iu2, v1)) ≤0 for all (iu2, v1) ∈D and such that v1 ≤−(1 + u2 2)/2. Let p(z) = 1 + p1z + p2z2 + · · · be analytic in U such that (p(z), zp′(z)) ∈D for all z ∈U. If Re(Φ(p(z), zp′(z))) > 0 (z ∈U), then Re(p(z)) > 0 (z ∈U).

Lemma 1.2. ([2])Let α ≥1 and 0 ≤η < 1. Suppose also that Ψ(η) = (3 −η) + p 5η2 −18η + 17 2(1 −η) . If β ≥Ψ(η), then Eα,β is starlike function of order η.

Lemma 1.3. ( [11])Let α ≥1 and β ≥1. Then

Theorem 2.1. Let αj ≥1, 0 ≤ηj < 1, and βj ≥ (3 −ηj) + q 5η2 j −18ηj + 17 2(1 −ηj) , for all j = 1, 2, 3, . . . , n. Suppose also that λ1, λ2, . . . , λn, ζ are positive real numbers such that n X j=1 1 −ηj λj

Corollary 2.2. Let α ≥1 and β ≥3+ √ 17 2 . Then Fα,β,λ,ζ(z) =  ζ zR 0 tζ−1  Eα,β(t) t 1/λ dt 1/ζ

Corollary 2.3. Let α ≥1 and β ≥3+ √ 17 2 . Then Fα,β,1,1(z) = zR 0  Eα,β(t) t  dt is starlike of order 1/2 in U. Example 2.4. Let E2,4(z) = 6[sinh √z−√z]/√z, then zR

Theorem 2.5. Let α1, α2, . . . , αn ≥1, β1, β2, . . . , βn ≥1 2(1 + √ 5) and consider the nor- malized Mittag-Leffler functions Eαj,βj defined by Eαj,βj(z) = Γ(βj)zEαj,βj(z). (2.9) Let β = min{β1, β2, . . . , βn} and λ1, λ2, . . . , λn be nonzero positive real numbers. Moreover, suppose that these numbers satisfy the following inequality 0 ≤1 − 2β + 1 β2 −β −1 n X j=1

Corollary 2.6. Let α ≥1, β ≥ 1 2(1 + √ 5) and λ > 0. Moreover, suppose that these numbers satisfy the following inequality 0 ≤1 − 2β + 1 λ(β2 −β −1) < 1. Then the function Fα,β,λ defined by Fα,β,λ(z) = z Z 0 Eα,β(t)