Research Article | Open Access
M. K. Aouf, A. O. Moustafa, E. A. Adwan, "Subclass of Multivalent Harmonic Functions Defi ned by Wright Generalized Hypergeometric Functions", Journal of Complex Analysis, vol. 2013, Article ID 397428, 7 pages, 2013. https://doi.org/10.1155/2013/397428
Subclass of Multivalent Harmonic Functions Defi ned by Wright Generalized Hypergeometric Functions
We introduce a new class of multivalent harmonic functions defi ned by Wright generalized hypergeometric function. Coefficient estimates, extreme points, distortion bounds, and convex combination for functions belonging to this class are obtained.
A continuous complex-valued function defined in a simply connected complex domain is said to be harmonic in if both and are real harmonic in . In any simply connected domain the function can be written in the form where and are analytic in , is called the analytic part, and the coanalytic part of . A necessary and sufficient condition for to be locally univalent and sense-preserving in is that in (see ).
Denote by the class of functions of the form (1) that are harmonic univalent and sense preserving in the unit disc for which .
Recently, Ahuja and Jahangiri  defined the class consisting of all -valent harmonic functions that are sense-preserving in and , are of the form
Let and be positive real parameters such that The Wright generalized hypergeometric function  (see also ) is defined by If and , we have the relationship where is the generalized hypergeometric function (see ) and By using the generalized hypergeometric function, Dziok and Srivastava  introduced a linear operator. Dziok and Raina in  and Aouf and Dziok in  extended this linear operator by using Wright generalized hypergeometric function.
Aouf et al.  defined the linear operator by the Hadamard product as where is given by We observe that, for a function of the form (2), we have where is given by (7) and is defined by For convenience, we write Now we can define the modified Wright operator as follows: where
For , , where is the modified Wright generalized hypergeometric functions (see ).
We note that, for and , we obtain , where is the modified Dziok-Srivastava operator (see ).
For , and for all , let denote the family of harmonic -valent functions , where and are given by (2) and satisfying the analytic criterion
Let be the subclass of consisting of functions such that and are of the form
2. Coefficient Estimates
Theorem 1. Let be such that and are given by (2). Furthermore, let Then is orientation preserving in and .
Proof. The inequality is enough to show that is orientation preserving. Note that Now we will show that . We only need to show that if (17) holds, then condition (15) is satisfied. Using the fact that if and only if , it suffices to show that where Substituting for and in (17) yields The last expression is nonnegative by (17). This completes the proof of Theorem 1. The harmonic -valent function where , shows that the coefficient bound given by (17) is sharp. It is worthy to note that the function of the form (22) belongs to the class for all because coefficient inequality (17) holds.
Theorem 2. A function is in the class , if and only if
Proof. From (15), we have the necessary and sufficient condition for that is, Hence we have the equivalent condition , . Simple algebraic manipulation of (25) yields This completes the proof of Theorem 2.
Theorem 3. A function , where and are given by (16), is in the class , if and only if
Proof . Since , we only need to prove the “only if” part of this theorem. To this end, for functions , where and given by (16), we notice that condition is equivalent to The above condition must hold for all , . Choosing the values of on the positive real axis where , we must have If condition (27) does not hold, then the numerator in (30) is negative for sufficiently close to . Hence there exist in for which the quotient in (30) is negative. This contradicts the required condition for . This completes the proof of Theorem 3.
3. Distortion Theorem
Theorem 4. Let the function , where and are given by (16), belong to the class . Then for , we have for . The results are sharp with equality for the functions defined by
Proof. We only prove the first inequality. The proof for the second inequality is similar and will be omitted. Let . Taking the absolute value of , we have This completes the proof of Theorem 4.
Corollary 5. Let the function , where and are given by (16), belong to the class . Then, for , we have for . The results are sharp with equality for the functions defined by where with and .
4. Extreme Points
Theorem 6. Let , where and are given by (16). Then , if and only if where , , , . In particular, the extreme points of the class are and .
Now we show that the class is closed under convex combinations of its members.
Theorem 7. The class is closed under convex combination.
Proof. For , let , where is given by Then by using Theorem 3, we have For , , the convex combination of may be written as Then by (45), we have This is the required condition and so . This completes the proof of Theorem 7.
The authors would like to thank the referees of the paper for their helpful suggestions.
- J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae A, vol. 9, pp. 3–25, 1984.
- O. P. Ahuja and J. M. Jahangiri, “Multivalent harmonic starlike functions,” Annales Universitatis Mariae Curie-Skłodowska, vol. 55, pp. 1–13, 2001.
- E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,” Proceedings of the London Mathematical Society, vol. 46, pp. 389–408, 1946.
- H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood, Chichester, UK, 1985.
- J. Dziok and H. M. Srivastava, “Classes of analytic functions associated with the generalized hypergeometric function,” Applied Mathematics and Computation, vol. 103, no. 1, pp. 1–13, 1999.
- J. Dziok and R. K. Raina, “Families of analytic functions associated with the Wright generalized hypergeometric function,” Demonstratio Mathematica, vol. 37, no. 3, pp. 533–542, 2004.
- M. K. Aouf and J. Dziok, “Certain class of analytic functions associated with the Wright generalized hypergeometric function,” Journal of Mathematics and Applications, vol. 30, pp. 23–32, 2008.
- M. K. Aouf, A. Shamandy, A. O. Mostafa, and S. M. Madian, “Certain class of -valent functions associated with the Wright generalized hypergeometric function,” Demonstratio Mathematica, vol. 43, no. 1, pp. 40–54, 2010.
- G. Murugusundaramoorthy and R. K. Raina, “On a subclass of harmonic functions associated with Wright's generalized hypergeometric functions,” Hacettepe Journal of Mathematics and Statistics, vol. 38, no. 2, pp. 129–136, 2009.
- R. Omar and S. A. Halim, “Multivalent harmonic functions defined by Dziok-Srivastava operator,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 35, no. 3, pp. 601–610, 2012.
- R. A. Al-Khal, “Goodman-Rønning-type harmonic univalent functions based on Dziok-Srivastava operator,” Applied Mathematical Sciences, vol. 5, no. 9–12, pp. 573–584, 2011.
- H. A. Al-Kharsani and R. A. Al-Khal, “Univalent harmonic functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 2, article 59, pp. 1–8, 2007.
- J. M. Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and Applications, vol. 235, no. 2, pp. 470–477, 1999.
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