Radii of starlikeness and convexity of $q-$Mittag--Leffler functions

Lemma 1.1. [7, p. 220] Let 0 ≤γ < 2. Then 1. If q satisfies the condition q−1(1 −q)(1 −qγ+1)(1 −qγ+2) > 1, γ ∈(0, 2), γ ̸= 1 then the zeros of z 7→Eγ,σ(z2; q) are all real, simple, symmetric and its positive zeros lie in the intervals, for n ∈N εγ,σ,n(q) ∈       

Lemma 1.2. Let σ is a fixed positive real number, 0 ≤γ < 2. Moreover, under the conditions of

Lemma 1.1 the function z 7→Eγ,σ(z2; q) has infinitely many zeros which are all real. Denoting by εγ,σ,n(q) the nth positive zero of z 7→Eν,c(z2; q), under the same conditions the Weierstrassian decomposition (1.3) Eγ,σ(z2; q) = 1 Γq(γ + 1) Y n≥1  1 − z2 ε2γ,σ,n(q)  is fulfilled, and this product is uniformly convergent on compact subsets of the complex plane.

Theorem 1.1. Let α ∈[0, 1) and with the conditions of Lemma 1.2 the following assertions hold true: a. The radius of starlikeness of order α of the function fγ,σ is r⋆ α(fγ,σ(z; q)) = xγ,σ,1(q), where xγ,σ,1(q) stands for the smallest positive zero of the equation rλ′ γ,σ(r; q) −(γ + 1)(α −1)λγ,σ(r; q) = 0. b. The radius of starlikeness of order α of the function gγ,σ is r⋆ α(gγ,σ(z; q)) = yγ,σ,1(q), where yγ,σ,1(q) stands for the smallest positive zero of the equation rλ′ γ,σ(r; q) −(α −1)λγ,σ(

Lemma 1.2 the q−Mittag-Leffler function has the infinite product representation given by Eγ,σ(z2; q) = 1 Γq(γ + 1) Y n≥1  1 − z2 ε2γ,σ,n(q)  and this infinite product is uniformly convergent on each compact subset of C. Taking into account fact that we use the notation λγ,σ(z; q) = Eγ,σ(z2; q), and by logarithmic differentiation we get λ′

Theorem 1.2. Let the conditions of Lemma 1.2 remain valid. a. The radius of starlikeness r⋆(fγ,σ(z; q)) satisfies the inequalities σ2(γ + 3)Γq(γ + 1) (γ + 1)Γq(γ + 3) −2σ2q2(γ + 5)Γq(γ + 3) (γ + 3)Γq(γ + 5) < (r⋆(fγ,σ(z; q)))−2 < σ2(γ + 3)Γq(γ + 1) (γ + 1)Γq(γ + 3) . b. The radius of starlikeness r⋆(gγ,σ(z; q)) satisfies the inequalities Γq(γ + 3) 3σ2Γq(γ + 1) < (r⋆(gγ,σ(z; q)))2 < 3Γq(γ + 3)Γq(γ + 5) σ2 9Γq(γ + 1)Γq(γ + 5) −10q2Γ2q(γ + 3)
. c. The radius of starlikeness r⋆(hγ,σ(z; q)) satisfies

Theorem 1.3. Let α ∈[0, 1) and with the conditions of Lemma 1.2 the following assrtions are valid: a. The radius of convexity rc α (fγ,σ(z; q)) is the smallest positive root of the transcendental equation (rfγ,σ(z; q))′ = αf ′ γ,σ(z; q). b. The radius of convexity rc α (gγ,σ(z; q)) is the smallest positive root of the transcendental equation (rgγ,σ(z; q))′ = αg′ γ,σ(z; q). c. The radius of convexity rc α (hγ,σ(z; q)) is the smallest positive root of the transcendental equation

Lemma 1.2. Observe also that limrց0 uγ,σ(r; q) = 1 and limrրξγ,σ,1 = −∞, which means that for z ∈Dr1 we get Re  1 + zf ′′ γ,σ(z; q) f ′γ,σ(z; q)  > α if and only if r1 is unique root of 1 + zf ′′ γ,σ(r; q) f ′γ,σ(r; q) = α situated in (0, ξγ,σ,1) . b. By virtue of (1.14) we have

Theorem 1.4. With the same conditions of Lemma 1.2 the following inequalities are valid: a. The radius of convexity rc (gγ,σ(z; q)) satisfies the inequalities Γq(γ + 3) 9σ2Γq(γ + 1) < (rc (gγ,σ(z; q)))2 < 9Γq(γ + 3)Γq(γ + 5) σ2 81Γq(γ + 1)Γq(γ + 5) −50q2Γ2q(γ + 3)
. b. The radius of convexity rc (hγ,σ(z; q)) satisfies the inequalities Γq(γ + 3) 4σ2Γq(γ + 1) < rc(hγ,σ(z; q)) < 2Γq(γ + 3)Γq(γ + 5) σ2 8Γq(γ + 1)Γq(γ + 5) −9q2Γ2q(γ + 3)
.

Corollary 1.1. Let α ∈[0, 1) and with the conditions of Lemma 1.2 the following assertions hold true: a. The radius of starlikeness of order α of the function f0,σ(z; q) = g0,σ(z; q) = z cos(q−1 2σz; q) is r⋆ α(f0,σ(z; q)) = x0,σ,1(q), where x0,σ,1(q) stands for the smallest positive zero of the equation rq−1 2 σ sin(q−1 2σz; q) + (α −1) cos(q−1 2 σz; q) = 0. b. The radius of starlikeness of order α of the function h0,σ = z cos(q−1 2σ√z; q) is r⋆ α(h0,σ(z; q)) = z0,σ,1(q), where z0,σ,1 stands fo

Corollary 1.2. Let the conditions of Lemma 1.2 remain valid. a. The radii of starlikeness r⋆(f0,σ(z; q)) and r⋆(g0,σ(z; q)) satisfies the inequalities 1 + q 3σ2 < (r⋆(f0,σ(z; q)))2 < 3(1 + q)(1 + q2)(1 + q + q2) σ2(9q4 + 9q3 + 8q2 + 9q + 9). b. The radius of starlikeness r⋆(hγ,σ(z; q)) satisfies the inequalities 1 + q 2σ2 < r⋆(hγ,σ(z; q)) < (1 + q)(1 + q2)(1 + q + q2) σ2(2q4 + 2q3 + q2 + 2q + 2). Setting γ = 0 in Theorem 1.3 we get the following results.

Corollary 1.3. With the same conditions of Lemma 1.2 the following inequalities are valid: a. The radius of convexity rc (g0,σ(z; q)) satisfies the inequalities 1 + q 9σ2 < (rc (g0,σ(z; q)))2 < 9(1 + q)(1 + q2)(1 + q + q2) σ2(81q4 + 81q3 + 112q2 + 81q + 81). b. The radius of convexity rc (h0,σ(z; q)) satisfies the inequalities 1 + q 4σ2 < rc(h0,σ(z; q)) < 2(1 + q)(1 + q2)(1 + q + q2) σ2(8q4 + 8q3 + 7q2 + 8q + 8). References [1] Akta¸s ˙I., Baricz ´A., Bounds for radii of starlikeness of some q−Bes