Radii of starlikeness of some special functions

Theorem 1. Let µ ∈(−1, 1), µ ̸= 0. The following statements hold: a) If 0 ≤α < 1 and µ ∈ −1 2, 0
, then r∗ α(fµ) = xµ,α, where xµ,α is the smallest positive root of the equation z s′ µ−1 2 , 1 2 (z) −α  µ + 1 2

Theorem 2. Let |ν| < 1 2. The following assertions are true: a) If 0 ≤α < 1, then r∗ α(uν) = δν,α, where δν,α is the smallest positive root of the equation zH′ ν(z) −α(ν + 1)Hν(z) = 0. b) If 0 ≤α < 1, then r∗(vν) = ρν,α, where ρν,α is the smallest positive root of the equation zH′ ν(z) −(α + ν)Hν(z) = 0. c) If 0 ≤α < 1, then r∗ α(wν) = σν,α, where σν,α is the smallest positive root of the equation zH′ ν(z) −(2α + ν −1)Hν(z) = 0. It is worth mentioning that the starlikeness of hµ, when µ ∈(−1, 1)

Lemma 1. Let ϕk(z) = 1F2  1; µ −k + 2 2 , µ −k + 3 2 ; −z2 4  where z ∈C, µ ∈R and k ∈{0, 1, . . . } such that µ −k is not in {0, −1, . . .}. Then, ϕk is an entire function of order ρ = 1 and of exponential type τ = 1. Consequently, the Hadamard’s factorization of ϕk is of the form (2.1)

Lemma 2. Consider the power series f(x) = X n≥0 anxn and g(x) = X n≥0 bnxn, where an ∈R and bn > 0 for all n ≥0. Suppose that both series converge on (−r, r), for some r > 0. If the sequence {an/bn}n≥0 is increasing (decreasing), then the function x 7→f(x)/g(x) is increasing (decreasing) too on (0, r). The result remains true for the power series f(x) = X n≥0 anx2n and

Lemma 3. Let p(x) = 1−a1x+a2x2 −a3x3 +· · ·+(−1)nanxn = (1−x/x1) · · · (1−x/xn) be a hyperbolic polynomial with positive zeros 0 < x1 ≤x2 ≤· · · ≤xn, and normalized by p(0) = 1. Then, for any constant C, the polynomial q(x) = Cp(x) −x p′(x) is hyperbolic. Moreover, the smallest zero η1 belongs to the interval (0, x1) if and only if C < 0. The proof is straightforward; it suffices to apply Rolle’s theorem and then count the sign changes of the linear combination at the zeros of p. We refer to [BDR,

Theorem 3. Let µ ∈(−1, 1), µ ̸= 0, and c be a constant such that c < µ + 1 2. Then the functions z 7→zs′ µ−1 2 , 1 2 (z) −csµ−1 2 , 1 2 (z) can be represented in the form (2.2) µ(µ + 1)  z s′ µ−1 2 , 1 2 (z) −c sµ−1

Lemma 3 implies that all zeros of gn( ˜ψµ; ζ) are real and positive and that the smallest one precedes the first zero of gn( ˜ϕ0; ζ). In view of Theorem A, the latter conclusion immediately yields that ˜ψµ ∈LPI and that its first zero precedes the one of ˜ϕ0. Finally, the first statement of the theorem for µ ∈(0, 1) follows after we go back from ˜ψµ and ˜ϕ0 to ψµ and ϕ0 by setting ζ = −z2 4 . Now we prove (2.2) for the case when µ ∈(−1, 0). Observe that for µ ∈(0, 1) the function [BK,

Lemma 3] ϕ1(z) = X k≥0 1 µ+1 2
k µ+2 2
k  −z2

Lemma 3 implies that for µ ∈(0, 1) all zeros of gn( ˜ψµ−1; ζ) are real and positive and that the smallest one precedes the first zero of gn( ˜ϕ1; ζ). This fact, together with Theorem A, yields that ˜ψµ−1 ∈LPI and that its first zero precedes the one of ˜ϕ1. Consequently, the first statement of the theorem for µ ∈(−1, 0) follows after we go back from ˜ψµ−1 and ˜ϕ1 to ψµ−1 and ϕ1 by setting ζ = −z2 4 and substituting µ by µ + 1. In order to prove the corresponding statement for (2.3), we recall first

Lemma 2.1]) ξµ,n ∈(nπ, (n + 1)π) for all µ ∈(0, 1) and n ∈{1, 2, . . . }, which implies that ξµ,n > ξµ,1 > π > 1 for all µ ∈(0, 1) and n ≥2. On the other hand, it is known that [BKS] if z ∈C and β ∈R are such that β > |z|, then (3.2) |z| β −|z| ≥ℜ  z β −z  . Then the inequality |z|2 ξ2µ,n −|z|2 ≥ℜ