Radii of Starlikeness and Convexity of Bessel Functions

Theorem 2.1. For 0 < q < 1, let ξν,1(q) and ζν,1(q) denote the first positive zero of the q-Bessel functions J(2) ν (·; q) and J(3) ν (·; q), respectively. For s ∈{2, 3}, if ν > 0, the function f (s) ν (·; q) and for ν > −1, the functions g(s) ν (·; q) and h(s) ν (·; q) have their lemniscate starlike radii; r∗ L(f (s) ν ), r∗ L(g(s) ν ) and r∗ L(h(s)

Theorem 2.2. For the q-Bessel function J(2) ν (·; q) and J(3) ν (·; q) with 0 < q < 1, let ξ′ ν,1(q) and ζ′ ν,1(q) denote the first positive zeros of dJ(s) ν (z; q)/dz for s = 2, 3, respectively and αν,1(q) and γν,1(q) be the first positive zeros of z·dJ(s) ν (z; q)/dz+(1−ν)J(s) ν (z; q) for s = 2 and s = 3, respectively. Similarly, let βν,1(q) and δν,1(q) denote the first positive zeros of z · dJ(s) ν (z; q)/dz + (2 −ν)J(s) ν (z; q) with s = 2 and s = 3, respectively. For s ∈{2, 3},

Corollary 2.3. Let ξν,1 denote the first positive zero of the Bessel function of first kind Jν. If ν > 0, the function fν and if ν > −1, the functions gν and hν have their radii of lemniscate starlikeness, r∗ L(fν), r∗ L(gν) and r∗ L(hν), respectively to be unique positive root of the equations (respectively) r2(J′ ν(r))2 −4rνJ′ ν(r)J′ ν(r) + 2ν2(Jν(r))2 = 0, (rJ′ ν(r) −νJν(r))2 −2(rJ′ ν(r) −νJν(r))Jν(r) −(Jν(r))2 = 0 and

Corollary 2.4. Let ν > −1. For the Bessel function of first kind Jν, if ν > 0, then the lemniscate convex radius of the function fν; rc L(fν), is the unique positive root of the equation rJ′′ ν (r) J′ν(r) + 1 ν −1  rJ′ ν(r) Jν(r) 2 −2 rJ′′

Lemma 2.5. Let ϕk(z) = 1F2  1; µ −k + 2 2 , µ + k + 3 2 ; −z2 4  , where z ∈C, µ ∈R and k ∈{0, 1, 2, . . .} such that µ −k ̸∈{0, −1, . . .}. Then, ϕk is an entire function of order ρ = 1. Consequently, the Hadamard’s factorization of ϕk is of the form ϕk(z) =

Theorem 2.6. Let µ ∈(−1, 1)\{0}. Then the lemniscate starlike radius of the normalized Lommel function of first kind fµ; r∗ L(fµ) is given by the smallest positive root of the equation 1 µ + 1 2
2 rs′ µ−1 2 , 1 2(r) sµ−1 2, 1 2(r)

Theorem 2.7. For the Lommel function of first kind sµ−1 2 , 1 2, the lemniscate convex radius of the function fµ with µ ∈(−1/2, 1) \ {0} and that of the functions gµ and hµ with µ ∈(−1, 1)\{0}; rc L(fµ), rc L(gµ) and rc L(hµ) are the smallest positive root of the equations (respectively) rs′′ µ−1 2 , 1 2(r) s′ µ−1

Theorem 2.8. Let α1 denote the first positive zero of the normalized Legendre polynomial of odd degree, P2n−1. Then the lemniscate starlike radius of P2n−1; r∗ L(P2n−1), is the unique positive root of the equation rP′ 2n−1(r) P2n−1(r) 2 −4 rP′ 2n−1(r) P2n−1(r)  + 2 = 0 in (0, α1).

Theorem 2.9. Let α1 be the first positive zero of the normalized Legendre polynomial of odd degree P2n−1.The lemniscate convex radius of the function P2n−1; rc L(P2n−1), is the smallest positive root of the equation (2.30) rP′′ 2n−1(r) P′ 2n−1(r) 2 −2 rP′′ 2n−1(r) P′ 2n−1(r)

Theorem 3.1. For s ∈{2, 3}, let ξν,1(q) and ζν,1(q) denote the first positive zeros of the q-Bessel functions J(2) ν (·; q) and J(3) ν (·; q), respectively where 0 < q < 1. Suppose ν > −1. Then the Janowski starlike radii of f (s) ν (·; q) (with ν > 0), g(s) ν (·; q) and h(s) ν (·; q); r∗ A,B(f (s) ν ), r∗ A,B(g(s) ν ) and r∗ A,B(h(s)

Theorem 3.2. For the q-Bessel functions J(2) ν (·; q) and J(3) ν (·; q) with 0 < q < 1, let ξ′ ν,1(q) and ζ′ ν,1(q) be the first positive zeros of dJ(s) ν (z; q)/dz for s = 2, 3, respectively and αν,1(q) and γν,1(q) be the first positive zeros of z·dJ(s) ν (z; q)/dz+(1−ν)J(s) ν (z; q) for s = 2 and s = 3, respectively. Similarly, let βν,1(q) and δν,1(q) denote the first positive zeros of z · dJ(s) ν (z; q)/dz + (2 −ν)J(s) ν (z; q) with s = 2 and s = 3, respectively. For s ∈{2, 3}, if ν > 0, then th

Corollary 3.3. Let ξν,1 is the first positive zero of the function the Bessel function of first kind Jν. Suppose ν > −1. Then the Janowski starlike radii of the function fν (with ν > 0), gν and hν; r∗ A,B(fν), r∗ A,B(gν) and r∗ A,B(hν) are respectively the unique root of the equations (respectively) rJ′ ν(r) Jν(r) −ν+ν  A −B 1 + |B|  = 0, rJ′ ν(r)

Corollary 3.4. For the Bessel function of first kind Jν, let ξ′ ν,1, αν,1 and βν,1 are the first positive zeros of J′ ν(z), zJ′ ν(z) + (1 −ν)Jν(z) and zJ′ ν(z) + (2 −ν)Jν(z), respectively. If ν > 0, the Janowski convex radius of the function fν; rc A,B(fν), if ν > −1, that of the functions gν; rc A,B(gν) and hν; rc A,B(hν) are the smallest roots of the equations (respectively) rJ′′ ν (r) J′ν(r) + 1

Theorem 3.5. Let ξµ,1 be the first positive zero of the Lommel function of first kind sµ−1 2, 1 2. Suppose µ ∈(−1, 1) \ {0}. Then the Janowski starlike radii of the functions fµ, (with −1/2 < µ < 1, µ ̸= 0), gµ and hµ, r∗ A,B(fµ); r∗ A,B(gµ) and r∗ A,B(hµ), are the unique positive roots of the equations rs′ µ−1 2, 1 2(r) sµ−1 2, 1

Theorem 3.1, clearly the radii of Janowski starlikeness of the functions gµ and hµ are the unique positive roots of the equations (3.12) and (3.13), respectively in (0, ξµ,1). Thus, the theorem holds. □ Now for the Janowski convexity of the normalizations of Lommel function of first kind, following the same steps as done in proof of the Theorem 3.2 with the similar notations as mentioned before, we get the following:

Theorem 3.6. Suppose µ ∈(−1, 1) \ {0}. For Lommel function of first kind sµ−1 2, 1 2, let ξ′ µ,1, ξµ,1, γµ,1 and δµ,1 denote the first positive zeros of s′ µ−1 2 , 1 2, sµ−1 2, 1 2, g′ µ and h′ µ, respectively. If µ ∈(−1/2, 1) \ {0}, then the function fµ and if µ ∈(−1, 1) \ {0}, the functions gµ and hµ, their radii of Janowski convexity rc A,B(fµ), rc

Theorem 3.7. Let α1 denote the first positive zero of the normalized Legendre polynomial of odd degree P2n−1. The Janowski starlike radius for the function P2n−1(z); r∗ A,B(P2n−1), is the unique positive root of the equation rP′ 2n−1(r) P2n−1(r) −1 + A −B 1 + |B| = 0 in (0, α1). With the calculations as performed in proof of Theorem 2.9, for the function P2n−1, if |z| < α1, then

Theorem 3.8. Let α1 be the first positive zero of the normalized Legendre polynomial of odd degree P2n−1. The radius of Janowski convexity for the function P2n−1(z); rc A,B(P2n−1), is the smallest positive root of the equation rP′′ 2n−1(r) P′ 2n−1(r) + A −B 1 + |B| = 0 in (0, α1).