73/1-2 (2008), 155–178

Lemma 1.1 ([15]). Let E be a set in the complex plane C and ψ : C3×D 7→C a function, that satisfies the admissibility condition ψ(ρi, σ, µ + νi; z) /∈E, where z ∈D, ρ, σ, µ, ν ∈R with µ+σ ≤0 and σ ≤−(1+ρ2)/2. If h : D →C, which satis- fies h(0) = 1, is analytic and for all z ∈D we have ψ ¡ h(z), zh′(z), z2h′′(z); z ¢ ∈E, then Re h(z) > 0 for all z ∈D. In particular, if we only have ψ : C2 × D 7→C, the admissibility condition reduces to ψ(ρi, σ; z) /∈E for all z ∈D and ρ, σ ∈R with σ ≤−(1 + ρ2)/2.

Lemma 1.2 ([18]). If the function f(z) = z + a2z2 + . . . + anzn + . . . is analytic in D and in addition 1 ≥2a2 ≥. . . ≥nan ≥. . . ≥0 or 1 ≤2a2 ≤. . . ≤ nan ≤. . . ≤2, then f is close-to-convex with respect to the convex function z 7→ −log(1−z). Moreover, if the odd function g(z) = z+b3z3 +. . .+b2n−1z2n−1 +. . . is analytic in D and if 1 ≥3b3 ≥. . . ≥(2n + 1)b2n+1 ≥. . . ≥0 or 1 ≤3b3 ≤. . . ≤ (2n + 1)b2n+1 ≤. . . ≤2, then g is univalent in D. We note that, as Ponnusamy and Vuorinen [20] pointe

Lemma 1.3 ([11]). If the function f(z) = a1z+a2z2+. . .+anzn+. . ., where a1 = 1 and an ≥0 for all n ≥2, is analytic in D and if the sequences {nan}n≥1, {nan −(n + 1)an+1}n≥1 both are decreasing, then f is starlike in D. Moreover, if for the analytic function g(z) = b1 + b2z + . . . + bn+1zn + . . ., where b1 = 1 and bn ≥0 for all n ≥2, we have that {bn}n≥1 is a convex decreasing sequence, i.e., bn −2bn+1 + bn+2 ≥0 and bn −bn+1 ≥0 for all n ≥1, then Re[g(z)] > 1/2 for all z ∈D. It is important t

Proposition 2.14. If b, p, c ∈C such that 2p + b + 1 ̸= 0, −2, −4, . . . , and z ∈C, then for the generalized Bessel function of the first kind of order p the following recursive relations hold: (i) zwp−1(z) + czwp+1(z) = (2p + b −1)wp(z); (ii) zw′ p(z) + (p + b −1)wp(z) = zwp−1(z); (iii) zw′ p(z) + czwp+1(z) = pwp(z); (iv) [z−pwp(z)]′ = −cz−pwp+1(z); (v) 2(2p + b + 1)u′ p(z) = −cup+1(z).

Theorem 2.16. If b, c, p ∈R and κ = p + (b + 1)/2, then the functions wp and up satisfy the following properties: (i) If κ ≥|c|/4 + 1, then Re up(z) > 0 for all z ∈D; (ii) If κ ≥|c|/4 and c ̸= 0, then up is univalent in D; (iii) If κ ≥|c|/4 + 1/2 and c ̸= 0, then up is convex in D; (iv) If κ ≥|c|/4 + 3/2 and c ̸= 0, then z 7→zup(z) is starlike in D; (v) If κ ≥|c|/2 + 1 and c ̸= 0, then z 7→zup(z) is starlike of order 1/2 in D; (vi) If κ ≥|c|/2 + 1 and c ̸= 0, then z 7→z1−pwp(z) is starlike in D.

Lemma 1.1. Hence by Lemma 1.1 we conclude Re h(z) = Re up(z) > 0 for all z ∈D. (ii) When κ ≥|c|/4 and c ̸= 0, then the above result implies Re up+1(z) > 0 for all z ∈D. Using part (v) of Proposition 2.14 we conclude that Re · −4κ c u′ p(z) ¸ = Re up+1(z) > 0 for all z ∈D. This in turn implies that up is close-to-convex with respect to the function ϕ(z) = −(cz)/(4κ). Now, since every close to convex function is univalent, it follows that up is univalent in D. We note that the univalence of the fu

Corollary 2.22. Let Jp : D →C be defined by Jp(z) = 2pΓ(p+1)z−pJp(z). Then the following assertions are true: (i) If Re p ≥1/4, then Re Jp(z1/2) > 0 for all z ∈D; (ii) If Re p ≥−3/4, then z 7→Jp(z1/2) is univalent in D; (iii) If Re p ≥−1/4 + (Im p)2/6, then z 7→Jp(z1/2) is convex in D; (iv) If Re p ≥3/4 + (Im p)2/6, then z 7→zJp(z1/2) is starlike in D; (v) If Re p ≥1/2 + (Im p)2/4, then z 7→zJp(z1/2) is starlike of order 1/2 in D;

Corollary 2.23. The following assertions are true: (i) If Re p ≥1/4, then Re Ip(z1/2) > 0 for all z ∈D; (ii) If Re p ≥−3/4, then z 7→Ip(z1/2) is univalent in D; (iii) If Re p ≥−1/4 + (Im p)2/6, then z 7→Ip(z1/2) is convex in D; (iv) If Re p ≥3/4 + (Im p)2/6, then z 7→zIp(z1/2) is starlike in D; (v) If Re p ≥1/2 + (Im p)2/4, then z 7→zIp(z1/2) is starlike of order 1/2 in D; (vi) If Re p ≥1/2 + (Im p)2/4, then z 7→z1−pIp(z) is starlike in D.

Corollary 2.24. Let Sp : D →C be defined by Sp(z) =2pΓ(p + 3/2)z−pSp(z). Then the following assertions are true: (i) If Re p ≥−1/4, then Re Sp(z1/2) > 0 for all z ∈D; (ii) If Re p ≥−5/4, then z 7→Sp(z1/2) is univalent in D; (iii) If Re p ≥−3/4 + (Im p)2/6, then z 7→Sp(z1/2) is convex in D; (iv) If Re p ≥1/4 + (Im p)2/6, then z 7→zSp(z1/2) is starlike in D; (v) If Re p ≥(Im p)2/4, then z 7→zSp(z1/2) is starlike of order 1/2 in D; (vi) If Re p ≥(Im p)2/4, then z 7→z1−pSp(z) is starlike in D. 3. Con

Theorem 3.1. If 0 ≤α < 1/2 and b, p, c ∈R, then the following assertions are true: (i) If 4κ ≥(1 −α)(1 −2α)−1/2|c| + 1, then Re up(z) > α for all z ∈D; (ii) If 4κ ≥(1 −α)(1 −2α)−1/2|c| and c ̸= 0, then up is close-to-convex of order α in D.

Proposition 2.14 we conclude that Re ·µ −4κ c ¶ u′ p(z) ¸ = Re up+1(z) > α for all z ∈D, i.e. up is close-to-convex of order α in D with respect to the function ϕ(z) = −(cz)/(4κ). □

Theorem 3.3. If 0 ≤α < 1 and b, p, c ∈R such that c ̸= 0 and 4α2+ (|c| −6)α + 2 ≥0, then the functions wp and up have the following properties: (i) If 4(1 −α)κ ≥|c| + 2(1 −α)(1 −2α), then up is convex of order α in D;

Lemma 1.1 to prove that Re q(z) > 0 for all z ∈D. For z = x + iy ∈D (with x, y ∈R) and ρ, σ ∈R satisfying σ ≤−(1 + ρ2)/2, we obtain Re ψ(ρi, σ; x + iy) = 4(1 −α)σ −4(1 −α)2ρ2 + cx −2(1 −α)e2 ≤−2(1 −α)(3 −2α)ρ2 + cx −2(1 −α)(1 + e2) < |c| + 2(1 −α)(1 −2α) −4(1 −α)κ ≤0. By Lemma 1.1 we conclude that Re q(z) > 0 for all z ∈D. This result implies Re · 1 + zu′′ p(z) u′p(z) ¸ = (1 −α) Re q(z) + α > α for all z ∈D, which shows that up is convex of order α in D.

Theorem 4.1. If c < 0 and b, p ∈R, then up has the following properties:

Lemma 1.2 it follows that f is close-to-convex with respect to the convex function −log(1 −z). (ii) Set g(z) = zup(z2) = z+b3z3 +. . .+b2n−1z2n−1 +. . ., where bn is defined by (2.11). Therefore we have 3b1 = −(3c)/(4κ) ≤1 and b2n−1 > 0 for all n ≥2. We want to show that {(2n −1)b2n−1}n≥2 is a decreasing sequence. Fix n ≥2. Then we have (2n −1)b2n−1 −(2n + 1)b2n+1 = b2n−1 · U2(n) 4n(κ + n −1), where U2(n) = 8n3 + 8(κ −3/2)n2 −4(κ −c/2 −1)n + c. Using the inequalities n3 ≥3n2 −3n + 1 and n2 ≥2n −1

Theorem 4.2. Let b, p be arbitrary real numbers and let c < 0. If 4κ ≥ −(c + 2) + p c2/2 −4c + 4, then the Alexander transform of the function z 7→ zup(z) is close-to-convex with respect to the function −log(1−z) and it is starlike in D. Moreover, we have that Re up(z) > 1/2 holds for all z ∈D.

Corollary 4.5. If b, p are arbitrary real numbers and c < 0 such that κ ≥−c/4 −1 and κ ̸= 0, then the function up is univalent in D.