Radii of starlikeness and convexity of generalized Mittag-Leffler functions

Lemma 1. If 1 α, β
∈Wi and γ > 0, then the function z 7→φ(α, β, γ, −z2) has infinitely many zeros which are all real. Denoting by λα,β,γ,n the nth positive zero of z 7→φ(α, β, γ, −z2), under the same conditions the Weierstrassian decomposition φ(α, β, γ, −z2) = 1 Γ(β) Y n≥1

Theorem 1. Let 1 α, β
∈Wi, γ > 0 and ρ ∈[0, 1). a. The radius of starlikeness of order ρ of fα,β,γ is r∗ ρ(fα,β,γ) = xα,β,γ,1, where xα,β,γ,1 is the smallest positive zero of the transcendental equation rλ′(α, β, γ, r) −β(ρ −1)λ(α, β, γ, r) = 0. b. The radius of starlikeness of order ρ of gα,β,γ is r∗ ρ(gα,β,γ) = yα,β,γ,1, where yα,β,γ,1 is the smallest positive zero of the transcendental equation rλ′(α, β, γ, r) −(ρ −1)λ(α, β, γ, r) = 0. c. The radius of starlikeness of order ρ of hα,β,γ is

Theorem 2. Let 1 α, β
∈Wi and γ > 0. a. The radius of starlikeness r∗(fα,β,γ) satisfies the inequalities γ(β + 2)Γ(β) βΓ(α + β) −(γ + 1)(β + 4)Γ(α + β) (β + 2)Γ(2α + β) < (r∗(fα,β,γ))−2 < γ(β + 2)Γ(β) βΓ(α + β) . b. The radius of starlikeness r∗(gα,β,γ) satisfies the inequalities 3γΓ(β) Γ(α + β) −5(γ + 1)Γ(α + β)

Theorem 3. Let 1 α, β
∈Wi, γ > 0 and ρ ∈[0, 1). a. The radius of convexity rc ρ(fα,β,γ) is the smallest positive root of the transcendental equation (rfα,β,γ(r))′ = ρf ′ α,β,γ(r). b. The radius of convexity rc ρ(gα,β,γ) is the smallest positive root of the transcendental equation (rgα,β,γ(r))′ = ρg′ α,β,γ(r). c. The radius of convexity rc ρ(hα,β,γ) is the smallest positive root of the transcendental equation

Theorem 4. Let 1 α, β
∈Wi and γ > 0. a. The radius of convexity rc(gα,β,γ) satisfies the inequalities 9γΓ(β) Γ(α + β) −25(γ + 1)Γ(α + β) 9Γ(2α + β) < (rc(gα,β,γ))−2 < 9γΓ(β) Γ(α + β). b. The radius of convexity rc(hα,β,γ) satisfies the inequalities 4γΓ(β) Γ(α + β) −9(γ + 1)Γ(α + β)

Lemma 1 they are interlaced with the zeros of z 7→Ψ(α, β, γ, z). Thus, Ψ′(α, β, γ, z) can be written as (2.6) Ψ′(α, β, γ, z) = β Γ(β)zβ−1 Y n≥1