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Abstract
For a fixed $a \in \{1, 2, 3, \ldots\},$ the radius of starlikeness of positive order is obtained for each of the normalized analytic functions \begin{align*} \mathtt{f}_{a, ν}(z)&:= \bigg(2^{a ν-a+1} a^{-\frac{a(aν-a+1)}{2}} Γ(a ν+1) {}_a\mathtt{B}_{2a-1, a ν-a+1, 1}(a^{a/2} z)\bigg)^{\tfrac{1}{a ν-a+1}},\\ \mathtt{g}_{a, ν}(z)&:= 2^{a ν-a+1} a^{-\frac{a}{2}(aν-a+1)} Γ(a ν+1) z^{a-aν} {}_a\mathtt{B}_{2a-1, a ν-a+1, 1}(a^{a/2} z),\\ \mathtt{h}_{a, ν}(z)&:= 2^{a ν-a+1} a^{-\frac{a}{2}(aν-a+1)} Γ(
Results & Lemmas (17)
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Proposition 1.1.
Proposition 1.1. [2, Proposition 2.2] Let a ∈N, and b, p, c, ∈R. Then aBb,p,c(z) = (2π) a−1 2 a−p−b 2 z 2 p a Y j=1 z 2aa/2 −p+j−1
Proposition 1.1. [2, Proposition 2.2] Let a ∈N, and b, p, c, ∈R. Then aBb,p,c(z) = (2π) a−1 2 a−p−b 2 z 2 p a Y j=1 z 2aa/2 −p+j−1
Proposition 1.2. · radius
Proposition 1.2. [2, Proposition 2.3] Let a ∈N, b, p, c ∈R and z ∈D. Then z a aBb,p−1,c(z) + c z 2 1−a zaBb,p+a,c(z) = 2p+b−1 a …
Proposition 1.2. [2, Proposition 2.3] Let a ∈N, b, p, c ∈R and z ∈D. Then z a aBb,p−1,c(z) + c z 2 1−a zaBb,p+a,c(z) = 2p+b−1 a aBb,p,c(z). 2. Radius of starlikeness of generalized Bessel functions The following preliminary result sheds insights into the zeros of the three functions
Theorem 2.1.
Theorem 2.1. Let ν > (a −1)/a, a ∈N. Then all zeros of aB2a−1,aν−a+1,1(aa/2z) are real. Further the origin is the only zero of…
Theorem 2.1. Let ν > (a −1)/a, a ∈N. Then all zeros of aB2a−1,aν−a+1,1(aa/2z) are real. Further the origin is the only zero of aB2a−1,aν−a+1,1(aa/2z) in the unit disk D.
Theorem 2.1
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Theorem 2.1 shows that the function fa,ν(z) = z
Theorem 2.1 shows that the function fa,ν(z) = z
Lemma 2.2.
Lemma 2.2. Let a ∈N and ν > −1/a. Then z aB′ 2a−1,aν−a+1,1(aa/2z) aB2a−1,aν−a+1,1(aa/2z) = zJν−1 (z) Jν (z) −(2 −a)ν + 1 −a.
Lemma 2.2. Let a ∈N and ν > −1/a. Then z aB′ 2a−1,aν−a+1,1(aa/2z) aB2a−1,aν−a+1,1(aa/2z) = zJν−1 (z) Jν (z) −(2 −a)ν + 1 −a.
Proposition 2.3.
Proposition 2.3. Let α, ν ∈R satisfy −1 < ν < −α. Then the equation rI′ ν(r) + αIν(r) = 0 has a unique root in (0, ∞).
Proposition 2.3. Let α, ν ∈R satisfy −1 < ν < −α. Then the equation rI′ ν(r) + αIν(r) = 0 has a unique root in (0, ∞).
Lemma 2.4.
Lemma 2.4. [12, p. 482] If ν > −1 and α, γ ∈R, then the Dini function z 7→αJν(z)+ γzJ′ ν(z) has all its zeros real whenever ((α/γ) + ν) ≥0.…
Lemma 2.4. [12, p. 482] If ν > −1 and α, γ ∈R, then the Dini function z 7→αJν(z)+ γzJ′ ν(z) has all its zeros real whenever ((α/γ) + ν) ≥0. In the case ((α/γ) + ν) < 0, it also has two purely imaginary zeros.
Lemma 2.5.
Lemma 2.5. [9, Theorem 6.1] Let α ∈R, ν > −1 and ν + α > 0. Further let xν,1 be the smallest positive root of αJν(z) + zJ′ ν(z) = 0. Then…
Lemma 2.5. [9, Theorem 6.1] Let α ∈R, ν > −1 and ν + α > 0. Further let xν,1 be the smallest positive root of αJν(z) + zJ′ ν(z) = 0. Then x2 ν,1 < j2 ν,1.
Lemma 2.6. · radius
Lemma 2.6. [8, p. 78] Let −1 < ν < −α, and ±iζ be the single pair of conjugate purely imaginary zeros of the Dini function z 7→αJν(z) + zJ′…
Lemma 2.6. [8, p. 78] Let −1 < ν < −α, and ±iζ be the single pair of conjugate purely imaginary zeros of the Dini function z 7→αJν(z) + zJ′ ν(z). Then ζ2 < − α + ν 2 + α + ν j2 ν,1. We are now ready to present the radius of starlikeness for each function given in (3).
Theorem 2.7.
Theorem 2.7. Let 0 ≤β < 1, and a ∈N. If ν > (a −1)/a, then r∗ β(fa,ν) = ja,f ν,β,1, where ja,f ν,β,1 is the smallest positive root of the…
Theorem 2.7. Let 0 ≤β < 1, and a ∈N. If ν > (a −1)/a, then r∗ β(fa,ν) = ja,f ν,β,1, where ja,f ν,β,1 is the smallest positive root of the equation raa/2J′ ν(r) − (ν −1)(1 −a)aa/2 + β(aν −a + 1) Jν(r) = 0. (6) If ν ∈(−1/a, (a −1)/a) and (aν −a + 1) aa/2 −β 2aa/2 + (aν −a + 1)
Proposition 2.3
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Proposition 2.3 that the root ia,f ν,β is unique. Finally, that ia,f ν,β < jν,n is a consequence of Lemma 2.6 and assumption (7). Indeed, …
Proposition 2.3 that the root ia,f ν,β is unique. Finally, that ia,f ν,β < jν,n is a consequence of Lemma 2.6 and assumption (7). Indeed, ia,f ν,β 2 < − (aν −a + 1) aa/2 −β 2aa/2 + (aν −a + 1) aa/2 −β j2
Corollary 2.8. · radius
Corollary 2.8. [4, Theorem 1(a)] Let 0 ≤β < 1. If ν > 0, then r∗ β(f1,ν) is the smallest positive root j1,f ν,β,1 of the equation rJ′ ν(r)…
Corollary 2.8. [4, Theorem 1(a)] Let 0 ≤β < 1. If ν > 0, then r∗ β(f1,ν) is the smallest positive root j1,f ν,β,1 of the equation rJ′ ν(r) −βνJν(r) = 0. In the case ν ∈(−1, 0), then r∗ β(f1,ν) is the unique positive root i1,f ν,β of the equation rI′ ν(r) −βνIν(r) = 0. The next two results find the radius of starlikeness of order β for the functions ga,ν and ha,ν given in (3).
Theorem 2.9.
Theorem 2.9. Let β ∈[0, 1), a ∈N, and ν > −1/a. If a(ν−1)(aa/2−1)+aa/2−β ≥0, then r∗ β(ga,ν) = ja,g ν,β,1, where ja,g ν,β,1 is the smallest…
Theorem 2.9. Let β ∈[0, 1), a ∈N, and ν > −1/a. If a(ν−1)(aa/2−1)+aa/2−β ≥0, then r∗ β(ga,ν) = ja,g ν,β,1, where ja,g ν,β,1 is the smallest positive root of the equation raa/2J′ ν(r) − (ν −1)(1 −a)aa/2 −a(1 −ν) + β Jν(r) = 0. (12)
Theorem 2.10.
Theorem 2.10. Let β ∈[0, 1), a ∈N, and ν > −1/a. If (aa/2 −1)(1−a+aν)+2(1− β) > 0, then r∗ β(ha,ν) = ja,h ν,β,1, where ja,h ν,β,1 is the…
Theorem 2.10. Let β ∈[0, 1), a ∈N, and ν > −1/a. If (aa/2 −1)(1−a+aν)+2(1− β) > 0, then r∗ β(ha,ν) = ja,h ν,β,1, where ja,h ν,β,1 is the smallest positive root of the equation aa/2rJ′ ν(r) + (aa/2 −1)(1 −a + aν) −aa/2ν + 2(1 −β) Jν(r) = 0. (14)
Theorem 2.10
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Theorem 2.10 when (aa/2 −1)(1 −a + aν) + 2(1 −β) < 0. 10
Theorem 2.10 when (aa/2 −1)(1 −a + aν) + 2(1 −β) < 0. 10
Theorem 3.1.
Theorem 3.1. For a fixed a ∈N, the function fa,ν given by (3) is starlike of order β ∈[0, 1) in D if and only if ν ≥νf(a, β), where νf(a, β)…
Theorem 3.1. For a fixed a ∈N, the function fa,ν given by (3) is starlike of order β ∈[0, 1) in D if and only if ν ≥νf(a, β), where νf(a, β) is the unique root of (aν −a + 1)(aa/2 −β)Jν(1) = aa/2Jν+1(1) in ((a −1)/a, ∞).
Theorem 3.2.
Theorem 3.2. Let a ∈N, ν > −1/a, and jν,1 be the first positive zero of Jν. Then the function ga,ν given by (3) is starlike of order β ∈[0,…
Theorem 3.2. Let a ∈N, ν > −1/a, and jν,1 be the first positive zero of Jν. Then the function ga,ν given by (3) is starlike of order β ∈[0, 1) in D if and only if ν ≥νg(a, β), where νg(a, β) is the unique root in (max{˜ν, −1/a}, ∞) of (a(ν −1)(aa/2 −1) + aa/2 −β)Jν(1) = aa/2Jν+1(1), and ˜ν ≃−0.7745 . . . is the unique root of jν,1 = 1. 12
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