MALAYA JOURNAL OF MATEMATIK

Lemma 2.1 ([13]). Suppose that f ∈A. If

Lemma 2.2 ([14]). Suppose that f ∈A and |f ′(ρ) −1| < 1 (2.3) is satisfied for each ρ ∈B, then f is convex in B 1 2 .

Lemma 2.3 ([23]). Let f ∈A. i. If

Theorem 3.1. Let ν > −1, k > 0, ν + k > 0, c ∈R and |c| < 8(ν + k). If the condition 8(ν + k) −3 |c| > 0 (3.1) hold, then gk ν,c(ρ) is univalent and starlike in B 1 2 .

Corollary 3.2. a. If k = c = 1 and ν > −5 8, then ρ 7→2νΓ(ν + 1)ρ1−νJν(ρ) is univalent and starlike in B 1 2 . b. If k = −c = 1 and ν > −5 8, then ρ 7→2νΓ(ν + 1)ρ1−νIν(ρ) is univalent and starlike in B 1 2 . In Theorem 3.1, taking certain special values of the parameters we deduce the following examples: Example 3.3. Let k = |c| = 1. i. If ν = 1 2, then the functions g1 1

Theorem 3.4. Let ν > −1, k > 0, ν + k > 0, c ∈R and |c| < 8(ν + k). If the condition 64(ν + k)2 −32 |c| (ν + k) + 3c2 > 0 (3.2) hold, then gk ν,c(ρ) is convex in B 1 2 =  ρ : |ρ| < 1 2

Corollary 3.5. If k = |c| = 1 and ν ∈(−1, ∞)\(−7 8, −5 8), then both functions g1 ν,1(ρ) = 2νΓ(ν+1)ρ1−νJν(ρ) and g1 ν,−1(ρ) = 2νΓ(ν + 1)ρ1−νIν(ρ) are convex in B 1 2 . Example 3.6. Assume that k = |c| = 1. i. If ν = 1 2, then the functions g1 1 2 ,1(ρ) and g1 1 2 ,−1(ρ) are convex in B 1 2 .

Theorem 3.7. Let ν > −1, k > 0, ν + k > 0, c ∈R and |c| < 8(ν + k). a. If the condition 64(ν + k)2 + 11c2 −256(ν + k) |c| 128(ν + k)2 −128(ν + k) |c| + 6c2 > 0 (3.3) hold, then the function gk ν,c(ρ) ∈UCV . b. If the condition 32 |c| (ν + k) [8(ν + k) −|c|] [8(ν + k) −|3c|] < 1 2 (3.4) hold, then the function gk ν,c(ρ) is in the class Sp. 20

Corollary 3.8. Suppose that k = |c| = 1. i. If 64ν2 −128ν −181 128ν2 + 128ν + 6 > 0, then both functions g1 ν,1(ρ) = 2νΓ(ν + 1)ρ1−νJν(ρ) and g1 ν,−1(ρ) = 2νΓ(ν + 1)ρ1−νIν(ρ) are in the class UCV . 22