On the characterization properties of certain hypergeometric functions in the open unit disk

Our purpose in the present investigation is to study certain geometric properties such as the close-to-convexity, convexity, and starlikeness of the hypergeometric function z1F2(a;b,c;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$z{}_{1}F_{2}(a;b,c;z)$\end{document} in the open unit disk. The usefulness of the main results for some familiar special functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the F10(−;c;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}_{0}F_{1}(-;c;z)$\end{document} function are also mentioned. We further consider a boundedness property of the function F21(a;b,c;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$_{1}F_{2}(a;b,c;z)$\end{document} in the Hardy space of analytic functions. Several corollaries and special cases of the main results are also pointed out.


Introduction and preliminaries
It is well known that special functions play important roles in geometric function theory, especially after the solution of the famous Bieberbach conjecture by De-Branges [10]. There exists an extensive literature that deals with the geometric properties of various special functions such as hypergeometric functions, confluent hypergeometric functons, Bessel's functions, Mittag-Leffler's function, Wright's function, Sturve's function, Lommel's function, Dini's function, and several other functions. Many researchers have determined sufficient conditions for the parameters involved in these special functions when they belong to certain classes of functions that are starlike, convex or close-to-convex. For instance, several sufficient conditions for the Gauss hypergeometric functions to be starlike or convex have been studied by Merkes and Scott [20], Lewis [19], Ruscheweyh and Singh [30], Miller and Mocanu [22], Silverman [32], Ponnusamy and Vuorinen [25], Küstner [17,18], and Hästo et al. [16]. Most of the known results in this direction deal with the shifted hypergeometric function z 2 F 1 (a, b; c; z) for real parameters a, b, and c. Recently, several authors have investigated the geometric properties of Bessel's functions [4,5,7,8], Struve's functions [23,36], Lommel's functions [9], Wright's function [26] (see also [13,28]), and Mittag-Leffler's function [3].
The classical generalized hypergeometric function [2,33] is defined by p F q (a 1 , a 2 , . . . , a p ; b 1 , b 2 , . . . , b q ; z) = ∞ n=0 (a 1 ) n (a 2 ) n · · · (a p ) n (b 1 ) n (b 2 ) n · · · (b q ) n z n n! , where the Pochhammer (or shifted factorial) symbols (b i ) n , i = 1, 2, . . . , q are assumed to be nonzero and nonnegative integers. In this paper, our main aim is to study various several geometric properties of the function z 1 F 2 (a; b, c; z), which is a particular case of the generalized hypergeometric function p F q (z) (for p = 1 and q = 2) and is defined by where both the parameters b and c do not assume values of zero or a negative integer. By the ratio test, we see that the radius of convergence of this series (1) is infinity, and hence it is an entire function. However, in this paper we consider this function in the restricted domain D = {z ∈ C : |z| < 1}, which is a suitable domain to be considered for studying its geometric properties. It can easily be verified that and further, the function w(z) = 1 F 2 (a; b, c; z) is a solution of the third-order homogeneous differential equation Now, we briefly outline some of the useful notations. Let H denote the class of analytic functions defined on D and let the subclass A = {f ∈ H : f (0) = 0 = f (0) -1} consist of analytic functions in D having a Taylor-series expansion of the form Evidently, the function z 1 F 2 (a; b, c; z) belongs to the class A. Our purpose in the present investigation is to study the geometric properties of the function z 1 F 2 (a; b, c; z) that enables us to deduce the corresponding properties for functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the 0 F 1 (-; c; z) function. We briefly give here the details of the special functions that stem from the function 1 F 2 (a; b, c; z).

The modified Struve function
The well-known Struve function of order ν is defined by which is a particular solution of the nonhomogeneous Bessel differential equation defined by The modified Struve function L ν (z) is defined by (see [37, p. 353]) The function L ν (z) does not belong to the class A, so we use the normalized form L ν (z) defined by It is easy to see that

The modified Lommel function
The Lommel function of the first kind S μ,ν (z) is a particular solution of the nonhomogeneous Bessel differential equation (see for details [6] and [35]) and can be expressed in terms of a hypergeometric series where μ ± ν is a nonnegative odd integer. The modified Lommel function is defined by For more details on the modified Lommel functions, we refer to the works of [29] and Shafer [31]. The function T μ,ν (z) obviously does not belong to the class A, so we use the normalized form T μ,ν (z) defined by which has a series representation of the form: ( 4 )

The modified Bessel function
For a = b in (1), we obtain which is a solution of the second-order homogeneous differential equation defined by It may be noted that a relationship exists between the function 0 F 1 and the modified Bessel function. Indeed, we have where the modified Bessel function is defined by and I ν (z) is the normalized modified Bessel function.
In order to have this paper reasonably self-contained, we give necessary details related to the consideration of geometric properties of functions analytic in the unit disk. We denote by S, the class of all functions f ∈ A that are univalent in D, that is, Recall the Bieberbach conjecture in which the Taylor coefficients of each function of the class S satisfy the inequality |a n | ≤ n for n = 2, 3, . . . , and only the rotations of the Koebe function z/(1z) 2 provide the case of equality. In 1984, De-Branges [10] settled the Bieberbach conjecture by using the generalized hypergeometric functions. The exploitation of hypergeometric functions in the proof of the Bieberbach conjecture has provided new areas of interest to study various special functions from the viewpoint of geometric function theory. We need the following basic classes of functions in the present investigation. For β < 1, let A function f ∈ S is called starlike (with respect to the origin 0), denoted by f ∈ S * if tw ∈ f (D) for all w ∈ f (D) and t ∈ [0, 1]. A function f ∈ S that maps D onto a convex domain is called a convex function and the class of such functions is denoted by K. For a given 0 ≤ α < 1, a function f ∈ S is called a starlike function of order α, denoted by S * (α), if and only if For a given 0 ≤ α < 1, a function f ∈ S is called a convex function of order α, denoted by K(α), if and only if It is well known that S * (0) = S * and K(0) = K. We recall [12] that the function zg (z) is starlike if and only if the function g(z) is convex. A function f ∈ S is said to be convex in the direction of the imaginary axis if f (D) has a connected intersection with every line parallel to the imaginary axis. Given a convex function g ∈ K with g(z) = 0 and α < 1, a function f ∈ S, is called close-to-convex of order α with respect to the convex function g, denoted by C g (α), if and only if The class C g (0) is the class of functions close-to-convex with respect to g. Geometrically, a function f ∈ S belongs to C if the complement E of the image-region F = {f (z) : |z| < 1} is the union of rays that are disjoint (except that the origin of one ray may lie on another one of the rays). The Noshiro-Warschawski theorem asserts that close-to-convex functions are univalent in D, but not necessarily the converse. It is easy to verify that K ⊂ S * ⊂ C.
For more details, one may refer to see [12].

Key lemmas
In order to establish our main results, we need the following results.
It may be noted that each convex decreasing sequence also generates a convex null sequence. We recall that the sequence a 0 , a 1 , . . . of nonnegative numbers is called a convex null sequence if lim k→∞ a k = 0 and a 0a 1 ≥ a 1a 2 ≥ · · · ≥ a ka k+1 ≥ · · · ≥ 0. For a convex null sequence a 0 = 1, a 1 , a 2 , . . . , we have instead of (6) the following inequality Re a 0 2 + ∞ n=1 a n z n > 0 (z ∈ D).
be a sequence of nonnegative real numbers such that A 1 = 1 and Then, the functions defined by the series n k=1 A k z k and ∞ n=1 A n z n are convex in the direction of the imaginary axis (see for details [1, Theorem 2.3.5, p. 34]).
If p(z) is analytic in D, with p(0) = 1 and ψ(p(z), zp (z), z 2 p (z); z) ∈ for z ∈ D, then This lemma is a special case of Theorem 1 due to Miller and Mocanu in [21].
In this paper, we study certain geometric properties such as the close-to-convexity, starlikeness, and convexity of the function z 1 F 2 (a; b, c; z). We also study the boundedness property of the function z 1 F 2 (a; b, c; z) in the concluding section. Several special cases and corollaries of our main results are also pointed out. z 1 F 2 (a; b, c; z) This section deals with conditions on the parameters a, b, and c so that the normalized function z 1 F 2 (a; b, c; z) is close-to-convex and hence univalent in D.

Close-to-convexity of
Theorem 1 Let a, b, c > 0 and a ≤ bc/2, then the function z 1 F 2 (a; b, c; z) is close-to-convex with respect to the convex function -log (1z).
Thus, we have X(n) ≥ 0 for all n ≥ 1, provided that a, b, c > 0 and a ≤ bc/2, and so the sequence {nA n } is nonincreasing. Applying Lemma 3, we conclude that the function z 1 F 2 (a; b, c; z) is close-to-convex with respect to the convex function -log(1z).
Example 1 Setting a = b in Theorem 1, it follows that for c ≥ 2, the function z 0 F 1 (-; c; z) is close-to-convex with respect to the convex function -log(1z) in D.
Proof Differentiating (1) with respect z, we obtain Using (8) and Theorem 1, we deduce that bc a z 1 F 2 (a; b, c; z) is close-to-convex with respect to -log(1z) and hence the function z 1 F 2 (a; b, c; z) is univalent in D.

Starlikeness of z 1 F 2 (a; b, c; z)
In this section we determine the conditions on the parameters a, b, and c such that the function z 1 F 2 (a; b, c; z) is not only close-to-convex with respect to -log(1z) but also starlike in D. Let KS * denote the family of functions in A that are close-to-convex with respect to -log(1z) and also starlike in D.
As before, if we set a = b in Theorem 3, then we deduce the following result: Proof Let g(z) = 1 F 2 (a; b, c; z) and f (z) = z 1 F 2 (a + 1; b + 1, c + 1; z). Then, from relation (8), we have f (z) = (bc/a)zg (z). By taking logarithmic derivatives of both sides, we obtain the relation Using Theorem 3 and the hypothesis of Theorem 4, we infer that f (z) is starlike and hence from (10), the function g(z)) = 1 F 2 (a; b, c; z) is a convex function.
Theorem 5 Let a, b, c > 0 and a ≤ bc/4, then z 1 F 2 (a; b, c; z) is convex in the direction of the imaginary axis.
This completes the proof.
Proof Let w(z) = 0 F 1 (-; c; z). If we put then p(z) is analytic in D and p(0) = 1. Therefore, to prove Theorem 6, we need to show that Re(p(z)) > 0 in D. Since the function w(z) = 0 F 1 (-; c; z) satisfies the differential equation we find that We may rewrite the above differential equation in the form of ψ(p(z), zp (z); z) = 0. Now, for all real s and t ≤ -(1 + s 2 )/2 and z(= x + iy) ∈ D, we have Also, for all x ∈ (-1, 1) and c satisfying the condition (11). Hence, we deduce that Re{ψ(is, t; z)} < 0, and therefore by Lemma 6, it follows that Re(p(z)) > 0 in D, which shows that the function 0 F 1 (-; c; z) is convex of order β.
If we use the identity then from Theorem 6, we obtain the following result:

Corollary 4 Let c be a real number such that
If we let f (z) = z 0 F 1 (-; c; z) and h(z) = f (z 2 /4)/(z/4), then we have This observation and Corollary 4 immediately yields the following result.
In view of bc ≥ a, it is easy to see that N(n) ≥ 0 for all n ≥ 2, and hence applying Lemma 1, we obtain the desired result.
If f ∈ R, then the convolution z 1 F 2 (a; b, c; z) * f (z) is in H ∞ ∩ R.
Applying Theorem 7, we have Re{ 1 F 2 (a; b, c; z)} > 1/2. Also, f (z) ∈ R implies that Re(f (z)) > 0, and therefore from Lemma 7 it follows that g ∈ R. Thus, in view of the first implication of (17), we have g ∈ H q for all q < 1. Further, from the second implication of (17), we have g ∈ H q/(1-q) for all 0 < q < 1, or equivalently, g ∈ H p for all 0 < p < ∞.
In view of the known bound for the Carathéodory functions in the unit disc [12], we note that if f (z) = z + ∞ n=2 a n z n ∈ R, then |a n | ≤ 2 n (n ≥ 2).
By using this bound, we have in view of (18): Using the well-known bound for the Carathéodory functions in the unit disc, we find that if f (z) = z + ∞ n=2 a n z n ∈ R then |a n | ≤ 2 n (n ≥ 2).
Using this fact we find that g(z) ≤ |z| + Therefore, applying Raabe's test for convergence, we deduce that the series ∞ n=1 (a) n (b) n (c) n n! 1 n (21) converges absolutely for |z| = 1. This argument shows that the power series for g(z) converges absolutely for |z| = 1. Furthermore, it is well known that [11, Theorem 3.11, p. 42] g ∈ H q implies continuity of g on D, the closure of D. Finally, since the continuous function g on the compact set D is bounded, g(z) is a bounded analytic function in D. Therefore, g ∈ H ∞ and this completes the proof.
We conclude this paper by considering a special case of Theorem 8. Indeed, if we put a = b in Theorem 8, we have the following corollary.