Radii of k-starlikeness of order α of Struve and Lommel functions

Theorem 1. Let |ν| ≤1 2, 0 ≤α < 1 and k ≥0. Then, the following assertions are true: i. The radius ru is the radius of k−starlikeness of order α of the normalized Struve function z 7→uν and it is the smallest positive root of the equation r(1+k)H′ ν(r)−(k +α)(ν+1)Hν(r) = 0 (2.1) in (0,hν,1), where hν,1 is the first positive zero of Struve function Hν. ii. The radius rv is the radius of k−starlikeness of order α of the normalized Struve function z 7→vν and it is the smallest positive root of the e

Theorem 2. The following assertions are true: i. Let µ ∈(−1 2,1) and µ ̸= 0. Then, the radius rf is the radius of k−starlikeness of order α of the normalized Lommel function z 7→fµ and it is the smallest positive root of the equation r(1+k)s′ µ−1 2 , 1 2 (r)−(k +α)(µ+ 1 2)sµ−1 2, 1 2 (r) = 0 (2.18) in (0,lµ,1), where lµ,1 is the first positive zero of Lommel function sµ−1 2, 1

Corollary 1. The following statements are true. i. The radius of k−starlikeness of order α of the function u 1 2 (z) =  2(1−cosz) √z  2 3 is the smallest positive root of the equation 2(1+k)rsinr +(1+4k +3α)(cosr −1) = 0 in  0,h 1 2 ,1 

Corollary 2. The following assertions are true. i. The radius of starlikeness of the function u 1 2 (z) =  2(1−cosz) √z  2 3 is r ∼= 2.7865 and it is the smallest positive root of the equation 2rsinr +cosr −1 = 0. ii. The radius of starlikeness of the function v 1 2 (z) = 2(1−cosz) z is r ∼= 2.33112 and it is the smallest positive root of the equation 2rsinr +2cosr −1 = 0. iii. The radius of starlikeness of the function w 1