On Geometric Properties of Normalized Hyper-Bessel Functions

In this paper, the normalized hyper-Bessel functions are studied. Certain sufficient conditions are determined such that the hyper-Bessel functions are close-to-convex, starlike and convex in the open unit disc. We also study the Hardy spaces of hyper-Bessel functions.


Introduction
Let H denote the class of functions that are analytic in U = {z : |z| < 1}, and A denote the class of functions f that are analytic in U having the Taylor series form The class S of univalent functions f is the class of those functions in A that are one-to-one in U .Let S * denote the class of all functions f such that f (U ) is star-shaped domain with respect to origin while C denotes the class of functions f such that f maps U onto a domain which is convex.A function f in A belongs to the class S * (α) of starlike functions of order α if and only if Re (z f (z) / f (z)) > α, α ∈ [0, 1) .For α ∈ [0, 1), a function f ∈ A is convex of order α if and only if Re (1 + z f (z) / f (z)) > α in U .This class of functions is dented by C (α) .It is clear that S * (0) = S * and C (0) = C are the usual classes of starlike and convex functions respectively.A function f in A is said to be close-to-convex function in U , if f (U ) is close-to-convex.That is, the complement of f (U ) can be expressed as the union of non-intersecting half-lines.In other words a function f in A is said to be close-to-convex if and only if Re (z f (z) /g (z)) > 0 for some starlike function g.In particular if g (z) = z, then Re ( f (z)) > 0. The class of close-to-convex functions is denoted by K.The functions in class K are univalent in U .For some details about these classes of functions one can refer to [1].Consider the class P δ (α) of functions p such that p (0) = 1 and Re e iδ p (z) These classes were introduced and investigated by Baricz [2].For δ = 0, we have the classes P 0 (α) and R 0 (α).Also for δ = 0 and α = 0, we have the classes P and R.
Special functions have great importance in pure and applied mathematics.The wide use of these functions have attracted many researchers to work on the different directions.Geometric properties of special functions such as Hypergeometric functions, Bessel functions, Struve functions, Mittag-Lefller functions, Wright functions and some other related functions is an ongoing part of research in geometric function theory.We refer for some geometric properties of these functions [2][3][4][5][6] and references therein.
We consider the hyper-Bessel function in the form of the hypergeometric functions defined as where the notation represents the generalized Hypergeometric functions and γ c represents the array of c parameters γ 1 , γ 2 , ..., γ c .By combining Equations ( 2) and (3), we get the following infinite representation of the hyper-Bessel functions since J γ c is not in class A. Therefore, consider the hyper-Bessel function J γ c which is defined by It is observed that the function J γ c defined in (5) is not in the class A. Here, we consider the following normalized form of the hyper-Bessel function for our own convenience.
For some details about the hyper-Bessel functions one can refer to [7][8][9].Recently Aktas et al. [8] studied some geometric properties of hyper-Bessel function.In particular, they studied radii of starlikeness, convexity, and uniform convexity of hyper-Bessel functions.Motivated by the above works, we study the geometric properties of hyper-Bessel function H γ c given by the power series (6).We determine the conditions on parameters that ensure the hyper-Bessel function to be starlike of order α, convex of order α, close-to-convex of order ( 1+α2 ).We also study the convexity and starlikeness in the domain U 1/2 = z : |z| < 1 2 .Sufficient conditions on univalency of an integral operator defined by hyper-Bessel function is also studied.We find the conditions on normalized hyper-Bessel function to belong to the Hardy space H p .
To prove our results, we require the following.
, where Proof.(i) By using the inequalities we obtain , where This implies that Furthermore, if we use the inequality By combining Equations (7) and (8), we obtain For H γ c ∈ S * (α) , we must have (ii) To prove that the function H γ c ∈ C (α) , we have to show that zH γc (z) By using the inequalities we have This implies that Furthermore, if we use the inequalities then we get By combining Equations (10) and (11) , we get This implies that H γ c ∈ C (α) , where 0 (iii) Using the inequality (10) and Lemma 4, we have . This shows that Therefore, Putting α = 0 in Theorem 1, we have the following results.
Consider the integral operator F β : U → C, where β ∈ C, β = 0, Here F β ∈ A. In the next theorem, we obtain the conditions so that F β is univalent in U .
then F β is univalent in U .

Proof. A calculations gives us
zF β (z) Since H γ c ∈ A, then by the Schwarz Lemma, triangle inequality and Equation (9) , we obtain This shows that the given integral operator satisfying the Becker's criterion for univalence [12], hence F β is univalent in U .

Uniformly Convexity of Hyper-Bessel Functions
.
By using Lemma 5, we have Hence, we obtain the required result.

Hardy Spaces Of Hyper-Bessel Functions
Let H ∞ denote the space of all bounded functions on H.