Research Article | Open Access
On Certain Sufficient Condition Involving Gaussian Hypergeometric Functions
The authors define a new subclass of of functions involving complex order in the open unit disk . For this new class, we obtain certain inclusion properties involving the Gaussian hypergeometric functions.
1. Introduction and Motivation
Let be the class of functions normalized by which are analytic in the open unit disk As usual, we denote by the subclass of consisting of functions which are also univalent in . A function is said to be starlike of order in if and only if
This function class is denoted by We also write where denotes the class of functions that are starlike in with respect to the origin.
A function is said to be convex of order in if and only if
The class of convex functions is denoted by the class Further, , the well-known standard class of convex functions. It is an established fact that
A function is said to be in the class of uniformly convex functions in if is a normalized convex function in and has the property that, for every circular arc contained in the unit disk , with center also in U, the image curve is a convex arc. The function class was introduced by Goodman .
For functions given by (1.1) and given by we define the Hadamard product (or Convolution) of and by
Furthermore, we denote by and two interesting subclasses of consisting, respectively, of functions which are -uniformly convex and -starlike in . Thus, we have The class was introduced by Kanas and Wiśniowska , where its geometric definition and connections with the conic domains were considered. The class was investigated in . In fact, it is related to the class by means of the well-known Alexander equivalence between the usual classes of convex and starlike functions; see also the work of Kanas and Srivastava  for further developments involving each of the classes and . In particular, when , we obtain where and are the familiar classes of uniformly convex functions and parabolic starlike functions in respectively (see for details, [1, 5]). In fact, by making use of a certain fractional calculus operator, Srivastava and Mishra  presented a systematic and unified study of the classes and .
A function is said to be in the class if it satisfies the inequality
The class was introduced earlier by Dixit and Pal . Two of the many interesting subclasses of the class are worthy of mention here. First of all, by setting the class reduces essentially to the class introduced and studied by Ponnusamy and Rønning , where Secondly, if we put we obtain the class of functions satisfying the inequality which was studied by (among others) Padmanabhan  and Caplinger and Causey .
Finally, many of the authors have also studied the class . For details of these works one can refer to the works of Ding Gong , R. Singh and S. Singh , Owa and Wu , and also the references cited by them. Although, many mapping properties of the class have been studied by these authors, they did not study any mapping properties involving the hypergeometric functions.
The Gaussian hypergeometric function , is given by is the solution of the homogeneous hypergeometric differential equation and has rich applications in various fields such as conformal mappings, quasiconformal theory, and continued fractions.
Here, , , are complex numbers such that , for , and for each positive integer , is the Pochhammer symbol. In the case of , , is defined if or where . In this situation, becomes a polynomial of degree in . Results regarding when is positive, zero, or negative are abundant in the literature. In particular when , the function is bounded. This and the zero balanced case are discussed in detail by many authors (see [14, 15]). The hypergeometric function has been studied extensively by various authors and it plays an important role in Geometric Function Theory. It is useful in unifying various functions by giving appropriate values to the parameters , and . We refer to [8, 16–19] and references therein for some important results.
In particular, the close-to-convexity (in turn the univalency), convexity, starlikeness, (for details on these technical terms we refer to ), and various other properties of these hypergeometric functions were examined based on the conditions on , and in . For more interesting properties of hypergeometric functions, one can also refer to [20, 21].
Let and be analytic in and univalent. Then we say that is subordinate to written as if and .
Using the integral representation, we can write
When equals the convex function , then the operator in this case becomes . For , , with then the convolution operator turns into Bernardi operator Indeed, and are known as Alexander and Libera operators, respectively.
Let and let be of the form (1.1). If , then the following coefficient inequalities hold true (cf. ): where is the coefficient of in the function which is the extremal function for the class related to the class by the range of the expression where is given, as above, by (1.22).
2. Properties of
Theorem 2.1. Let and be of the form (1.1). If , then The estimate is sharp.
Proof. Since , we have where is analytic in and satisfies the condition and for . Hence, we have Using and , we have By equating the coefficients, we observe that the coefficient in the right-hand side depends only on on the left-hand side of the above expression. This gives By using we get Squaring both sides of (2.6) and integrating around , we obtain By letting we conclude that or By making use of the fact that we get This gives The result is sharp for the function
Theorem 2.2. Let . Then a sufficient condition for is The result is sharp for the function
Proof. In view of (2.13), which is clearly less than or equal to zero for all , Letting we get Thus, .
3. Results Involving Gaussian Hypergeometric Function
Theorem 3.1. Let Also, let c be a real number such that . Then a sufficient condition for the function to be in the class is that where
Proof. has the series representation given by
In view of Theorem 2.2, it suffices to show that
From the fact that we observe that is real and positive, under the hypothesis By writing as, we get Using the fact that it is easy to see that From (1.14), By using the Gauss summation theorem we get Equation (3.4) now follows by an application of (3.1) and (3.2).
Theorem 3.2. Let . Also, let be a real number such that If and if the inequality is satisfied, then where
Proof. Let be of the form (1.1) belong to the class By virtue of Theorem 2.2, it suffices to show that Taking into account inequality (2.1) and the relation we deduce that which is bounded previously by in view of inequality (3.12).
Repeating the previous reasoning for we can improve the assertion of Theorem 3.2 as follows.
Theorem 3.3. Let . Also, let be a real number such that If and if the inequality is satisfied, then where .
In the special case when Theorem 3.2 immediately yields the following new result.
Theorem 3.4. Let . Also, let be a real number such that If and if the inequality is satisfied, then where .
Theorem 3.5. Let . Also, let be a real number such that If and if the inequality is satisfied, then
Proof. Let Applying the well-known estimate for the coefficients of the functions due to de Branges , we need to show that The left-hand side of (3.18) can be written as The second expression of (3.19), by virtue of the triangle inequality for the pochhammer symbol is less than or equal to Now, making use of the relation (3.7), we get where we are writing By repeating the use of (3.7) and the Gauss summation formula, we have As a next step, we consider the first expression of equation. By making use of the triangle inequality for the pochhammer symbol as stated in evaluating we get Now making use of relation (3.7), we obtain where we write By repeating the use of (3.7) and the Gauss summation formula, we have The proof of Theorem 3.5 now follows by an application of the inequalities of the terms dealing with and inequality (3.17).
Repeating the previous reasoning for we can improve the assertion of Theorem 3.5 as follows.
Theorem 3.6. Let . Also, let be a real number such that If and if the inequality is satisfied, then
In the special case when Theorem 3.2 immediately yields a result concerning the Carlson-Shaffer operator
Theorem 3.7. Let . Also, let be a real number such that If and if the inequality is satisfied, then
Theorem 3.8. Let . Also, let be a real number such that where is given with (1.22). If, for some (), and the inequality is satisfied, then
Proof. By means of (1.17) and (2.13), the following inequality must be satisfied:
Applying the estimates for the coefficients given by (1.21), and making use of the relations (3.7) and condition (3.29) will be satisfied if provided The proof of the Theorem 3.8 is now completed by virtue of hypothesis (3.28).
Theorem 3.9. Let . Also, let be a real number such that where is given with (1.22). If, for some and the inequality is satisfied, then
Proof. Proceeding as in the proof of Theorem 3.8, and applying the estimates for the coefficients given by (1.24) instead of (1.21), and making use of relations (3.7) and the proof of the theorem by virtue of hypothesis (3.31) is complete.
The authors sincerely thank the referees for their suggestions.
- A. W. Goodman, “On uniformly convex functions,” Annales Polonici Mathematici, vol. 56, no. 1, pp. 87–92, 1991.
- S. Kanas and A. Wiśniowska, “Conic regions and -uniform convexity,” Journal of Computational and Applied Mathematics, vol. 105, no. 1-2, pp. 327–336, 1999.
- S. Kanas and A. Wiśniowska, “Conic domains and starlike functions,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 45, no. 4, pp. 647–657, 2000.
- S. Kanas and H. M. Srivastava, “Linear operators associated with -uniformly convex functions,” Integral Transforms and Special Functions, vol. 9, no. 2, pp. 121–132, 2000.
- A. W. Goodman, Univalent Functions, Vols. I and II, Polygonal Publishing, Washington, NJ, USA, 1983.
- H. M. Srivastava and A. K. Mishra, “Applications of fractional calculus to parabolic starlike and uniformly convex functions,” Computers & Mathematics with Applications, vol. 39, no. 3-4, pp. 57–69, 2000.
- K. K. Dixit and S. K. Pal, “On a class of univalent functions related to complex order,” Indian Journal of Pure and Applied Mathematics, vol. 26, no. 9, pp. 889–896, 1995.
- S. Ponnusamy and F. Rønning, “Duality for Hadamard products applied to certain integral transforms,” Complex Variables Theory and Application, vol. 32, no. 3, pp. 263–287, 1997.
- K. S. Padmanabhan, “On a certain class of functions whose derivatives have a positive real part in the unit disc,” Annales Polonici Mathematici, vol. 23, pp. 73–81, 1970.
- T. R. Caplinger and W. M. Causey, “A class of univalent functions,” Proceedings of the American Mathematical Society, vol. 39, no. 2, pp. 357–361, 1973.
- Y. Ding Gong, “Properties of a class of analytic functions,” Mathematica Japonica, vol. 41, no. 2, pp. 371–381, 1995.
- R. Singh and S. Singh, “Convolution properties of a class of starlike functions,” Proceedings of the American Mathematical Society, vol. 106, no. 1, pp. 145–152, 1989.
- S. Owa and Z. Wu, “A note on certain subclass of analytic functions,” Mathematica Japonica, vol. 34, no. 3, pp. 413–416, 1989.
- R. Balasubramanian, S. Ponnusamy, and M. Vuorinen, “On hypergeometric functions and function spaces,” Journal of Computational and Applied Mathematics, vol. 139, no. 2, pp. 299–322, 2002.
- S. Ponnusamy, “Hypergeometric transforms of functions with derivative in a half plane,” Journal of Computational and Applied Mathematics, vol. 96, no. 1, pp. 35–49, 1998.
- B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,” SIAM Journal on Mathematical Analysis, vol. 15, no. 4, pp. 737–745, 1984.
- A. Gangadharan, T. N. Shanmugam, and H. M. Srivastava, “Generalized hypergeometric functions associated with -uniformly convex functions,” Computers & Mathematics with Applications, vol. 44, no. 12, pp. 1515–1526, 2002.
- Y. C. Kim and F. Rønning, “Integral transforms of certain subclasses of analytic functions,” Journal of Mathematical Analysis and Applications, vol. 258, no. 2, pp. 466–489, 2001.
- T. N. Shanmugam, “Hypergeometric functions in the geometric function theory,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 433–444, 2007.
- H. Silverman, “Convolutions of univalent functions with negative coefficients,” Annales Universitatis Mariae Curie-Skłodowska Section A, vol. 29, pp. 99–107, 1975.
- H. Silverman, “Starlike and convexity properties for hypergeometric functions,” Journal of Mathematical Analysis and Applications, vol. 172, no. 2, pp. 574–581, 1993.
- Ju. E. Hohlov, “Operators and operations on the class of univalent functions,” Izvestiya Vysshikh Uchebnykh Zavedeniĭ Matematika, no. 10, pp. 83–89, 1978.
- L. de Branges, “A proof of the Bieberbach conjecture,” Acta Mathematica, vol. 154, no. 1-2, pp. 137–152, 1985.
Copyright © 2009 H. Silverman et al. 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.