On Harmonic Meromorphic Functions Associated with Basic Hypergeometric Functions

By making use of basic hypergeometric functions, a class of complex harmonic meromorphic functions with positive coefficients is introduced. We obtain some properties such as coefficient inequality, growth theorems, and extreme points.


Introduction and Preliminaries
Recently, the popularity of the study of basic hypergeometric series (also called -hypergeometric series) is increasing among the researchers. Back in 1748, Euler considered the infinite product ( ; ) −1 ∞ = ∏ ∞ =0 (1 − +1 ) −1 and ever since it became very important to many areas. However, it was stated in the literature that the development of these functions was slower until Heine (1878) converted a simple observation such that lim → 1 [(1− )/(1− )] = , which then returns the theory of 2 1 basic hypergeometric series to the famous theory of Gauss 2 1 hypergeometric series. Various authors [1][2][3][4][5][6][7] introduced classes of analytic functions involving hypergeometric functions and investigated their properties. However, all these results concern mostly the Gaussian and generalized hypergeometric functions. It seems that no attempt has been made to derive similar results for the basic hypergeometric functions. In this work, we proceed to define a class of harmonic meromorphic functions in the unit disk associated with the basic hypergeometric function and discuss some of its properties.
The subject of harmonic univalent functions is a recent area of research which was initially established by Clunie and Sheil-Small [8]. The importance of these functions is due to their use in the study of minimal surfaces as well as in various problems related to applied mathematics and perhaps to other areas of sciences. Hengartner and Schober [9] have introduced and studied special classes of harmonic functions, which are defined on the exterior of the unit diskŨ = { : | | > 1}. They have proved that these functions are complex valued harmonic, sense preserving, univalent mappings , admitting the representation where ℎ( ) and ( ) are analytic inŨ = { : | | > 1}.
For ∈ U \ {0}, let denote the class of functions which are harmonic in the punctured unit disk U \ {0}, where ℎ( ) and ( ) are analytic in U \ {0} and U, respectively. The class was studied in [10][11][12]. We further denote by the subclass of consisting of functions of the form which are univalent harmonic in the punctured unit disk U \ {0}.
The -derivative of a function ℎ( ) is defined by For a function ℎ( ) = , observe that where ℎ ( ) is the ordinary derivative. For more properties of , see [14,15]. Now for ∈ U, 0 < | | < 1, and = + 1, the basic hypergeometric function defined in (7) where, for convenience, Corresponding to the function Φ([ 1 , ], ), defined in (12), consider and its inverse function The series expansion of the inverse function is given as where Now, let which is analytic function in the punctured unit disk U \ {0}.
] , The Scientific World Journal That is, The inequality in (28) holds true for all (0 ≤ < 1). Therefore, letting → 1 in (28), we obtain By hypothesis (27), it follows that (26) holds, so that ∈ ([ 1 , , ]). Note that is sense preserving in U \ {0}. This is because The Scientific World Journal 5 Since R( ) ≤ | | for all , it follows from (33) that We now choose the values of on the real axis. Upon the clearing the denominator in (34) and letting → 1 through real values, we obtain the following: which immediately yields the required condition (32).
Next, we consider a distortion property for functions in the class ([ 1 , , ]) as follows.
Proof. We will only prove the right-hand inequality. The proof for the left-hand inequality is similar and we will omit it: The functions for 0 ≤ < 1 show that the bounds given in Theorem 4 are sharp in U \ {0}.
Next, we give the following.
and note that, by Theorem 3, 0 ≥ 0. Consequently, we obtain This proves the theorem.

Remark 6.
Other work related to -hypergeometric functions and analytic functions can be found in [16,17].