Abstract
In the present paper, we find sufficient conditions for starlikeness and convexity of normalized Lommel functions of the first kind using the admissible function methods. Additionally, we investigate some inclusion relationships for various classes associated with the Lommel functions. The functions belonging to these classes are related to the starlike functions, convex functions, close-to-convex functions and quasi-convex functions.
Highlights
Let A denote the family of functions f of the form: ∞∑ an zn f (z) = z +n =2 which are analytic in the open unit disk D and satisfy the usual normalization condition f (0) = f 0 (0) − 1 = 0
N =2 which are analytic in the open unit disk D and satisfy the usual normalization condition f (0) = f 0 (0) − 1 = 0
C(α) denote the subclasses of A consisting of functions which are starlike of order α and convex of order α in D, respectively. These classes are characterized by the equivalence:
Summary
Let A denote the family of functions f of the form:. n =2 which are analytic in the open unit disk D and satisfy the usual normalization condition f (0) = f 0 (0) − 1 = 0. C(α) denote the subclasses of A consisting of functions which are starlike of order α and convex of order α in D, respectively. These classes are characterized by the equivalence:. Let S ∗ (0) = S ∗ and C(0) = C which are the classes of starlike functions and convex functions, respectively. In [14,15], various inclusion relationships associated with several subclasses of analytic functions were investigated. Motivated by their works, by using the linear operator Lμ,ν , we define new subclasses of A as follows: z( Lμ,ν f (z))0. Lemma 4. ([2] Yağmur) If μ > −1, ν ∈ R where μ ± ν are not negative odd integers, and (μ + 1)[(μ + 1)(μ + 3) − ν2 ] ≥ , R hμ,ν (z)/z > 0 in D
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