The radius of uniform convexity of Bessel functions

Theorem 2.1. Let f be a function of the form f(z) = z + P∞ n=2 anzn, and analytic in the disk U(r). The function f is uniformly convex in the disk U(r) if and only if (2.1) Re  1 + zf ′′(z) f ′(z)  >

Lemma 2.1. i. If a > b > r ≥|z|, and λ ∈[0, 1], then (2.2)

Theorem 3.1. If ν > 0, then the radius of uniform convexity of the function fν is the smallest positive root of the equation 1 + 2(r2 −ν2)Jν(r) rJ′ν(r) + 2  1 −1 ν  rJ′ ν(r) Jν(r) = 0. Moreover ruc(fν) < rc(fν) < j′ ν,1 < jν,1, where jν,1 and j′ ν,1 denote the first positive zeros of Jν and J′

Theorem 3.2. i. If ν > −1, then the radius of uniform convexity of the function gν is the smallest positive root of the equation 1 + 2r(2ν −1)Jν+1(r) −rJν(r) Jν(r) −rJν+1(r) = 0. Moreover, rc α(gν) < αν,1 < jν,1, where αν,1 is the first positive zero of the Dini function z 7→(1 −ν)Jν(z) + zJ′ ν(z). ii. If ν ∈(−2, −1), then the radius of uniform convexity of the function gν is ruc(gν), where ruc(gν) is the unique root of the equation 1 + 2rrIν(r) −(2ν −1)Iν+1(r) Iν(r) + rIν+1(r) = 0, in the interv

Theorem 3.3. i. If ν > −1, then the radius of uniform convexity of the function hν is the smallest positive root of the equation 1 + r 1 2 2(ν −1)Jν+1(r 1 2 ) −r 1 2Jν(r 1 2) 2Jν(r 1 2 ) −r 1

Theorem 3.4. The function fν is uniformly convex in U if and only if ν > ν1 ≃1.4426..., where ν1 is the unique root of the equation ν(3ν −2) (Jν(1))2 + ν(4ν −5)Jν(1)Jν−1(1) + 2(1 −ν) (Jν−1(1))2 = 0 situated in (ν∗, ∞), where ν∗≃0.39001... is the root of the equation J′ ν(1) = 0.

Theorem 3.5. The function gν is uniformly convex in U if and only if ν > ν2 ≃2.44314..., where ν2 is the unique root of the equation (4ν −3)Jν+1(1) −Jν(1) = 0 situated in [0, ∞).

Theorem 3.6. The function hν is uniformly convex in U if and only if ν > ν3 ≃0.30608..., where ν3 is the unique root of the equation (2ν −3)Jν+1(1) + Jν(1) = 0 situated in [0, ∞).