Research Article | Open Access

# On Certain Sufficient Condition Involving Gaussian Hypergeometric Functions

**Academic Editor:**Teodor BulboacΔ

#### Abstract

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 [1].

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 WiΕniowska [2], where its geometric definition and connections with the conic domains were considered. The class was investigated in [3]. 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 [4] 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 [6] 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 [7]. 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 RΓΈnning [8], where Secondly, if we put we obtain the class of functions satisfying the inequality which was studied by (among others) Padmanabhan [9] and Caplinger and Causey [10].

Finally, many of the authors have also studied the class . For details of these works one can refer to the works of Ding Gong [11], R. Singh and S. Singh [12], Owa and Wu [13], 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, 16β19] 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 [5]), and various other properties of these hypergeometric functions were examined based on the conditions on , and in [8]. 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 .

For , we recall that the operator of Hohlov [22] which maps into itself defined by where denotes usual Hadamard product of power series. Therefore, for a function defined by (1.1), we have

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. [2]): 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).

Similarly, if of the form (1.1) belong to the class , then (cf. [3]) 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 [23], 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.

#### Acknowledgment

The authors sincerely thank the referees for their suggestions.

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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.