Certain Properties of q-Hypergeometric Functions

Uzoamaka A. Ezeafulukwe and Maslina Darus 1Mathematics Department, Faculty of Physical Sciences, University of Nigeria, Nsukka, Nigeria 2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi, Selangor Darul Ehsan, Malaysia Correspondence should be addressed to Maslina Darus; maslina@ukm.edu.my Received 4 June 2015; Revised 3 August 2015; Accepted 3 August 2015 Academic Editor: Teodor Bulboaca Copyright © 2015 U. A. Ezeafulukwe and M. Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The quotients of certain q-hypergeometric functions are presented as g-fractions which converge uniformly in the unit disc. These results lead to the existence of certain q-hypergeometric functions in the class of either q-convex functions, PC q , or q-starlike functionsPS* q .


Introduction and Preliminaries
Let A  be the class of analytic functions  in the unit disc U = { :  ∈ C, || < 1}, normalized by (0) =   (0) − 1 = 0 and of the form In this paper, we extend some results obtained in the theory of functions to the -theory and to achieve this we write out some standard notations and basic definitions used in this paper.
In [4], Srivastava and Owa summarized some properties of functions that belong to the class PS ⋆  of -starlike functions in U, introduced and investigated by Ismail et al. in [5].Srivastava and Owa [4] further proposed the study of properties of functions that belong to the class PS ⋆  () of -starlike functions of order , 0 ≤  ≤ 1, and also of the functions that belong to the class PC  of -convex functions in U.The authors [4] also defined the class of functions PC  , on the function  ∈ A  , (0) =   (0) − 1 = 0 as follows: while [5] defined the class of functions PS ⋆  , on the function Meanwhile, Agrawal and Sahoo in [6] defined and studied some properties of functions that belong to the class PS ⋆  () and also Sahoo and Sharma in [7] (see also [8]) defined and studied the class PK  of -close-to-convex functions.Kanas and Rȃducanu in [9] also used the Ruscheweyh -differential operator to introduce and study some properties of (, ) uniformly starlike functions of order .Other works related to -hypergeometric can also be traced in [10][11][12][13].
In addition, in [14] Baricz and Swaminathan used the Alexander duality between starlike and convex function to define the following.They used Definition 4 to establish Theorem 5 as follows.
Theorem 5.If 0 < , , ,  < 1 satisfies the following two more conditions Apart from Theorem 5, to the best of our knowledge, other properties of functions in the class of functions PC  are yet to be studied.For more results and further studies on -calculus see [15][16][17].
Motivated by the numerous studies, of the abovementioned authors, we aimed, in this paper, at using the parameters , ,  as real parameters and placing some constraints on quotients of two or more hypergeometric functions to establish the following: We describe the procedures to achieve (⧫i) as follows: ( (iv) Calculate the continued fraction of the hypergeometric quotient on the right-hand side of the equation and convert the continued fractions to a -fraction which converges uniformly.
(v) Hence, the -fractions lead to the geometric properties of Pick functions.
(vi) Then substituting last result in the third outline with simple calculation on the outcome gives the required results.
The description (i) to (vi) can also be used to establish (⧫ii).First we write out the known results needed to establish ours.Ismail and Libis noted in [18] that the hypergeometric function satisfies the -difference equation They rewrite (13) in the form iterated the functional relationship (14), and got They also noted that if these iterations converge, this will give rise to the continued fraction where They also modified a result in [19] to Theorem 6.Later on, we will modify Theorem 6 to suit our results.
Theorem 6 (see [18]).Let || ̸ = 1 with exception of the zeros of 2 Φ 1 [, ; ; , ]; the continued fraction of a meromorphic function of , which is equal to the function 2 Φ 1 (, ; ; , )/ 2 Φ 1 (, ; ; , ), is represented by throughout the -plane with It is equal to this ratio in a neighborhood  at origin and furnishes the analytic continuation of it throughout the finite plane.
We review that the sequence of positive numbers, {  },  ≥ 0, with   = 1, is called a Hausdorff moment sequence if there exists a positive (Borel) measure  on the close interval [0, 1] such that Equation ( 20) can also be represented by with Θ analytic in the slit complex domain C\[1, ∞], and also Θ belongs to the set of Pick functions.Lemma 7 bridges the gaps between total monotone sequence, Hausdorff moment sequence, the set of Pick functions, and -fractions.
Theorem 8 (see [20]).Letting  1 ,  2 ,  3 , . . .be constant satisfying one of the following conditions: then the continued fraction converges uniformly for We write Remark 9, which is some comments made by Wall in [20].This remark enables us to convert our originally derived continued fraction into a -fraction which converges uniformly.
Remark 9.By means of equivalence transformation, the continued fraction of the form can be transformed into continued fraction of the form (25), by first reducing the partial denominator to unity; hence we obtain Also (10) We state necessary and sufficient condition for a function  to be in the class PS ⋆  established in [5].
Theorem 18.If the hypothesis of Theorem 17 holds and there exists an increasing function,   , mapping [0, 1] into itself, with Theorem 19.If the hypothesis of Theorem 18 holds, then

Proof of the Main Results
Proof of Theorem 12.To calculate the continued fraction of ,  2 ;  2 ; , ] , . . .
Remark 20.The addition of the leading constant  0 does not affect the convergence of the continued fraction 2 Φ 1 (, ; ; , )/ 2 Φ 1 (, ; ; , ).The continued fraction on the right-hand side of (65) converges to the left-hand side provided (, , ) belongs to the neighborhood of (0, 0, 0) with  ∈ U and  is not a pole of the right-hand side.We also note that the continued fraction (65) is called the infinite fraction.
Furthermore, since ∫ ( By hypothesis of Theorem 15 and  > 0, we need to show that 4 ≥  2 .Hence, Theorem 15 is established.
Proof of Theorem 16.The calculations in Theorem 16 were established in [18].
Proof of Theorem 17.We then need to reduce the partial denominators of (44) to unity, by setting The proofs of Theorems 18 and 19 are the same procedures as those of Theorems 14 and 15 and hence are omitted in this paper.