Geometric Properties of Lommel Functions of the First Kind

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.


Introduction
Let A denote the family of functions f of the form: which are analytic in the open unit disk D and satisfy the usual normalization condition f (0) = f (0) − 1 = 0. Let S denote the subclass of A which are univalent in D. Also let S * (α) and C(α) denote the subclasses of A consisting of functions which are starlike of order α and convex of order α in D, respectively.Analytically, these classes are characterized by the equivalence: For convenience, let S * (0) = S * and C(0) = C which are the classes of starlike functions and convex functions, respectively.Furthermore, let C(β, α) and C * (β, α) be the subclasses of A defined by respectively.The functions in the classes C(β, α) and C * (β, α) are known as close-to-convex functions and quasi-convex functions, respectively.The Lommel function of the first kind s µ,ν which is expressed in terms of a hypergeometric series where µ ± ν are not negative odd integers, is a particular solution of the following inhomogeneous Bessel differential equation [1]: It is observed that the function s µ,ν does not belong to the class A. Recently, Ya gmur [2] and Baricz et al. [3] considered the following function h µ,ν defined by: and they obtained some geometric properties of the function h µ,ν .For another interesting properties of Lommel function, we can refer to [4,5].
The above function h µ,ν belongs to A and is expressed by: where (λ) n is the Pochhammer symbol which defined in terms of Euler's gamma function such that (λ Corresponding to the function h µ,ν defined by (1), we consider a linear operator L µ,ν : A → A defined by: in terms of the Hadamard product (or convolution) * .Then it can be easily observed from (1) and ( 2) that the following relation holds: In a few years ago, many authors introduced new subclasses of univalent (or multivalent) functions by using several linear operators and found many properties of them [6][7][8][9][10][11][12][13].In [14,15], various inclusion relationships associated with several subclasses of analytic functions were investigated.
Motivated by their works, by using the linear operator L µ,ν , we define new subclasses of A as follows: Here, we note that a function f belongs to the class S * µ,ν (α) (K µ,ν (α), C µ,ν (β, α) and C * µ,ν (β, α)) is equivalent to that the function L µ,ν f (z) belongs to the class S * (α) (K(α), C(β, α) and C * (β, α), respectively).Further, from the linearity of the operator L µ,ν , the following relations hold: and In the present paper some geometric properties of the normalized Lommel function of the first kind are obtained by applying the method of admissible function.In Section 2, we find some sufficient conditions for starlikeness and convexity for the function h µ,ν .In Section 3, we investigate some inclusion relationships for the classes S * µ,ν (α), K µ,ν (α), C µ,ν (β, α) and C * µ,ν (β, α) which are related to the function h µ,ν .
The following lemmas will be used for the proof of our results.Lemma 1. ([16] Miller and Mocanu) Let Ω be a set in the complex plane C and let b be a complex number such that R(b) > 0. Suppose that the function ψ :

Lemma 2. ([17] Miller and Mocanu
Let p be an analytic function in D such that p(0) = 1 and (p(z), zp (z)) For analytic functions f and g, we say that f is subordinate to g, denoted by f ≺ g, if there is an analytic function then the definition of subordination f ≺ g can be simplified into the conditions f (0) = g(0) and f (D) ⊆ g(D) (See [18], p. 36).

Lemma 3. ([19] Eenigenburg et al.)
Let h be convex univalent in D and w be analytic in D with R {w(z)} ≥ 0 in D. If q is analytic in D and q(0) = h(0), then the subordination

Sufficient Conditions for Starlikeness and Convexity
We find some sufficient conditions for starlikeness and convexity of the function h µ,ν given by (1).
Theorem 1.Let µ and ν be real numbers such that µ ± ν are not negative odd integers, µ > 2, Then the function h µ,ν is a starlike univalent function in D.
Example 1.We note that µ = 5/2 and ν = 1/2 satisfy the condition of Theorem 1. Therefore the function is starlike in D.
Theorem 2. Let µ and ν be real numbers such that µ ± ν are not negative odd integers, µ > 2, and Then the function h µ,ν is a convex univalent function in D.
Proof.Let f ∈ S * µ,ν (α) and define a function φ : C → C by Then φ is analytic in D and φ(0) = 1.From the equality (3), we get By combining ( 21) and ( 22), we obtain Now, by applying the logarithmic differentiation on both sides of (23) and multiplying the resulting equation by z, we have which, in view of (21), yields Now, we define a function Φ : Observe that Φ is continuous on < 0 which shows that R {Φ(iu 2 , v 1 )} < 0. Therefore, by Lemma 2, we have Thus, by making use of (21), we find that f ∈ S * µ+1,ν+1 (α).This completes the proof of Theorem 3. Theorem 4. Let µ, ν and α be real numbers such that µ ± ν are not negative odd integers, 0 ≤ α < 1 and 2α Proof.By applying (4) and Theorem 3, we observe that which proves Theorem 4.