Honam Mathematical J. 42 (2020), No. 4, pp. 719–731

Lemma 1.1. [20] If the function f(z) = z + a2z2 + ... + anzn + ... is analytic in U and in addition 1 ≥2a2 ≥... ≥nan ≥... ≥0 or 1 ≤2a2 ≤... ≤nan... ≤2, then f(z) is close-to-convex function with respect to the convex function z →−Log (1 −z) .

Lemma 1.2. [20] If the odd function g(z) = z + b3z3 + ... + b2n−1z2n−1+... is analytic in U and if 1 ≥3b3 ≥... ≥(2n+1)b2n+1... ≥0 or 1 ≤3b3 ≤... ≤(2n + 1)b2n+1... ≤2, then g(z) is univalent in U. We can verify directly that if a function f : U →C satisfies the hypothesis of Lemma 1.1, then it is close-to-convex with respect to the convex function z 7→1 2 log 1 + z 1 −z 

Lemma 1.3. [18] A function f defined in (5) belongs to the classes S∗(α) and C (α) , if it satisfies the following conditions ∞ X n=2 (n −α) |an| ≤ 1 −α, α ∈[0, 1) , ∞ X n=2 n (n −α) |an| ≤ 1 −α,

Lemma 1.4. [6] If f ∈A satisfy |f′(z) −1| < 1 for each z ∈U, then f is convex in U1/2 =  z : |z| < 1 2

Lemma 1.5. [8] If f ∈A satisfy f(z) z −1 < 1 for each z ∈U, then f is starlike in U1/2 =  z : |z| < 1 2

Lemma 1.6. [9] If f ∈A satisfy |f′(z) −1| < 2 √ 5 for each z ∈U, then f is starlike in U1/2 =  z : |z| < 1 2

Lemma 1.7. [16] Let β ∈C with ℜ(β) > 0, c ∈C with |c| ≤1, c ̸= −1. If h ∈A satisfies c |z|2β +  1 −|z|2β zh′′(z) βh′(z) ≤1, z ∈U,

Lemma 1.8. [19] If f ∈A satisfies zf′′(z) f′(z) < 1 2, then f ∈UCV.

Lemma 1.9. [5] Let {an}∞ n=1 be a sequence of non negative real numbers such that a1 = 1. If {an}∞ n=2 is convex decreasing. i.e. 0 ≥ an+2 −an+1 ≥an+1 −an, then ℜ ( ∞ X n=1 anzn−1 ) > 1 2, (z ∈U) . 2. Close to Convexity of Modified Lommel Functions with

Theorem 2.1. If κ, η ∈R+ and κ ≥η with inequality B ≥ 3 8 (V + 1) −1, then z →L(z) is close-to-convex with respect to convex function −log (1 −z) .

Theorem 2.2. If κ, η ∈R+ and κ ≥η with inequality B ≥ 5 12 (V + 1) −1, then z →L(z) is close-to-convex with respect to convex function 1 2 log  1+z 1−z  .

Theorem 3.1. If κ, η ∈R+, α ∈[0, 1) and κ ≥η with inequality Φ {Φ (2 −α) + α −1} ≤4ΦBV (Φ −1) (1 −α) , where, Φ = (B + 1) (V + 1) , then L defined in (4) belongs to the class S∗(α) .

Theorem 3.2. If κ, η ∈R+, α ∈[0, 1) and κ ≥η with inequality Φ (2 −α) ≤2BV (1 −α) (Φ −1) , where, Φ = (B + 1) (V + 1) , then L defined in (4) belongs to the class C (α) .

Lemma 1.3 we need only show that ∞ X n=2 n (n −α) |an| ≤1 −α.

Theorem 3.3. If κ, η ∈R+ and κ ≥η with inequalities 4 (B+n −2) (V+n −2) ≥ 1, (B+n −2) (V+n −2) (B+n −1) (V+n −1) + 1 ≥ 8 (B+n −1) (V+n −1) , then ℜ L(z) z  > 1 2,

Theorem 4.1. Let κ, η ∈R+ and κ ≥η. Then the following asser- tions are true: (i) If (B+1)(V+1) 4(B+1)(V+1)−1 < BV, then Lκ,η is starlike in U1/2. (ii) If (B+1)(V+1) (B+1)(V+1)−1 < 2BV, then Lκ,η is convex in U1/2.

Theorem 4.2. Let α ∈[0, 1) , κ, η ∈R+ and κ ≥η, G =4 (B+1) (V+1) = (κ + 5)2 −η2 and H =4BV = (κ + 3)2 −η2. Then for all z ∈U the following assertions are true:

Theorem 4.3. Let κ, η ∈R+ and κ ≥η. Let (G−2) (GH −G −H) > G (G−1) and suppose that M is a positive real number such that |Lκ,η (z)| ≤M in the open unit disc. If |β −1| + G (G−1) (G−2) (GH −G −H) + M |β| ≤1, then Fβ is univalent in U.

Theorem 5.1. Let κ, η ∈R+ and κ ≥η. If (G−3) (2GH−4G−3H) > 4G (2G−3) , then Lκ,η ∈UCV.