Research Article | Open Access

G. Murugusundaramoorthy, N. Magesh, "Certain Subclasses of Starlike Functions of Complex Order Involving Generalized Hypergeometric Functions", *International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 178605, 12 pages, 2010. https://doi.org/10.1155/2010/178605

# Certain Subclasses of Starlike Functions of Complex Order Involving Generalized Hypergeometric Functions

**Academic Editor:**Stanisława R. Kanas

#### Abstract

Making use of the generalized hypergeometric functions, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients and obtain coefficient estimates, extreme points, the radii of close-to-convexity, starlikeness and convexity, and neighborhood results for the class . In particular, we obtain integral means inequalities for the function that belongs to the class in the unit disc.

#### 1. Introduction

Let denote the class of functions of the form

which are analytic in the open unit disc and normalized by Also denote by the subclass of consisting of functions of the form

introduced and studied by Silverman [1]. For functions given by (1.1) and given by it is known [2] the definition of the Hadamard product (or Convolution) of and by

For positive real values of and the *generalized hypergeometric function * is defined by

where denotes the set of all positive integers and is the Pochhammer symbol defined by

The notation is quite useful for representing many well-known functions such as the exponential, the Binomial, the Bessel, the Laguerre polynomial, and others; for example, see [3].

Let be a linear operator defined by

where

unless otherwise stated. For notational simplicity, we can use a shorter notation for in the sequel. The linear operator is called Dziok-Srivastava operator (see [3]), includ (as its special cases) various other linear operators introduced and studied by Carlson and Shaffer [4], Owa [5], Ruscheweyh [2], and Srivastava and Owa [6]. Motivated by Goodman [7, 8], Rønning [9, 10] introduced and studied the following subclasses of A function is said to be in the class , uniformly -starlike functions if it satisfies the condition

and is said to be in the class uniformly -convex functions if it satisfies the condition

Indeed it follows from (1.8) and (1.9) that

The interesting geometric properties of these function classes were extensively studied by Kanas et al., in [11–14]. Motivated by Altintas et al. [15], Murugusundaramoorthy and Srivastava [16], and Murugusundaramoorthy and Magesh [17], now, we define a new subclass uniformly starlike functions of complex order.

For , and , we let be the class of functions satisfying (1.2) with the analytic criterion

where is given by (1.6).

By suitably specializing the values of , , and the class , leads to various new subclasses of starlike functions of complex order. As for illustrations, we present some examples for the cases.

*Example 1.1. *If with , and of the form of (1.2), then

*Example 1.2. *If with , and of the form of (1.2), then
where is called Ruscheweyh derivative of order defined by

*Example 1.3. *If with , , and of the form of (1.2), then
where is a Bernardi operator [18] defined by

*Example 1.4. *If with , and of the form of (1.2), then
where is a well-known Carlson-Shaffer linear operator [4] defined by
Also was studied by Murugusundaramoorthy and Magesh [19].

The main object of this paper is to study some usual properties of the geometric function theory such as the coefficient bound, extreme points, radii of close to convexity, starlikeness, and convexity for the class Further, we obtain neighborhood results and integral means inequalities for aforementioned class.

#### 2. Basic Properties

First we obtain the necessary and sufficient condition for functions in the class

Theorem 2.1. *The necessary and sufficient condition for of the form of (1.2) to be in the class is
**
where , and *

*Proof. *Assume that , then
Letting along the real axis, we have
Hence, by maximum modulus theorem, the simple computational leads the desired inequality
Conversely, suppose that (2.1) is true for Then
if
That is if
which completes the proof.

Corollary 2.2. *Let the function defined by (1.2) belong to Then
** and, , with equality for
*

Next we state the following theorem on extreme points for the class without proof.

Theorem 2.3 (Extreme Points). *Let
**
Then if and only if can be expressed in the form , where and *

#### 3. Close-to-Convexity, Starlikeness, and Convexity

We determine the radii of close-to-convexity, starlikeness, and convexity results for functions in the class in the following theorems.

Theorem 3.1. *Let Then is close-to-convex of order in the disc , where
*

*Proof. *Let belong to It is known [20] that is close-to-convex of order , if it satisfies the condition
For the left-hand side of (3.2) we have
The last expression is less than if
Using the fact that if and only if
we can say that (3.2) is true if
Or, equivalently,
which completes the proof.

Theorem 3.2. *Let Then the following are given.** is starlike of order in the disc where** is convex of order in the unit disc , where**
Each of these results is sharp for the extremal function given by (2.10).*

*Proof. *Let It is known [1] that is starlike of order , if it satisfies the condition
For the left-hand side of (3.10) we have
The last expression is less than if
Using the fact that if and only if
we can say that (3.10) is true if
Or, equivalently,
which yields the starlikeness of the family.

Using the fact that is convex if and only if is starlike, we can prove (2), on lines similar to those the proof of (1).

#### 4. Integral Means

In order to find the integral means inequality and to verify the Silverman Conjuncture [21] for we need the following subordination result due to Littlewood [22].

Lemma 4.1 (see [22]). *If the functions and are analytic in with , then
*

Applying Theorem 2.1 with the extremal function and Lemma 4.1, we prove the following theorem.

Theorem 4.2. *Let If and is nondecreasing sequence, then, for and one has
**
where , and *

*Proof. *Let of the form of (1.2) and , then we must show that
By Lemma 4.1, it suffices to show that
Setting
from (4.5) and (2.1) we obtain
This completes the proof of Theorem 4.2.

#### 5. Inclusion Relations Involving

To study about the inclusion relations involving we need the following definitions due to Goodman [23] and Ruscheweyh [24]. The neighborhood of function is given by

Particularly for the identity function , we have

Theorem 5.1. *Let
**
where Then *

*Proof. *For , Theorem 2.1 yields
so that
On the other hand, from (2.1) and (5.5) we have

Now we determine the neighborhood for each of the class which we define as follows. A function is said to be in the class if there exists a function such that

Theorem 5.2. *If and
**
where then *

*Proof. *Suppose that , then we find from (5.6) that
which implies the coefficient inequality
Next, since , we have
so that
provided that is given precisely by (5.8). Thus by definition, for given by (5.8), which completes the proof.

#### Concluding Remarks

By suitably specializing the various parameters involved in Theorems 2.1 to 5.2, we can state the corresponding results for the new subclasses defined in Examples 1.1 to 1.4 and also for many relatively more familiar function classes.

#### Acknowledgment

The authors would like to record their sincerest thanks to the referee(s) for their valuable suggestions and comments.

#### References

- H. Silverman, “Univalent functions with negative coefficients,”
*Proceedings of the American Mathematical Society*, vol. 51, pp. 109–116, 1975. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Ruscheweyh, “New criteria for univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 49, pp. 109–115, 1975. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Dziok and H. M. Srivastava, “Certain subclasses of analytic functions associated with the generalized hypergeometric function,”
*Integral Transforms and Special Functions*, vol. 14, no. 1, pp. 7–18, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Owa, “On the distortion theorems. I,”
*Kyungpook Mathematical Journal*, vol. 18, no. 1, pp. 53–59, 1978. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. M. Srivastava and S. Owa, “Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions,”
*Nagoya Mathematical Journal*, vol. 106, pp. 1–28, 1987. View at: Google Scholar | MathSciNet - A. W. Goodman, “On uniformly convex functions,”
*Annales Polonici Mathematici*, vol. 56, no. 1, pp. 87–92, 1991. View at: Google Scholar | Zentralblatt MATH | MathSciNet - A. W. Goodman, “On uniformly starlike functions,”
*Journal of Mathematical Analysis and Applications*, vol. 155, no. 2, pp. 364–370, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Rønning, “Uniformly convex functions and a corresponding class of starlike functions,”
*Proceedings of the American Mathematical Society*, vol. 118, no. 1, pp. 189–196, 1993. View at: Publisher Site | Google Scholar | MathSciNet - F. Rønning, “Integral representations of bounded starlike functions,”
*Annales Polonici Mathematici*, vol. 60, no. 3, pp. 289–297, 1995. View at: Google Scholar | MathSciNet - S. Kanas and A. Wisniowska, “Conic regions and $k$-uniform convexity,”
*Journal of Computational and Applied Mathematics*, vol. 105, no. 1-2, pp. 327–336, 1999. View at: Publisher Site | Google Scholar | MathSciNet - S. Kanas and A. Wiśniowska, “Conic domains and starlike functions,”
*Revue Roumaine de Mathematiques Pures et Appliquees*, vol. 45, no. 4, pp. 647–657, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Kanas and H. M. Srivastava, “Linear operators associated with $k$-uniformly convex functions,”
*Integral Transforms and Special Functions*, vol. 9, no. 2, pp. 121–132, 2000. View at: Publisher Site | Google Scholar | MathSciNet - S. Kanas and T. Yaguchi, “Subclasses of $k$-uniformly convex and starlike functions defined by generalized derivative. II,”
*Publications Institut Mathematique*, vol. 69(83), pp. 91–100, 2001. View at: Google Scholar | MathSciNet - O. Altintas, O. Ozkan, and H. M. Srivastava, “Neighborhoods of a class of analytic functions with negative coefficients,”
*Applied Mathematics Letters*, vol. 13, no. 3, pp. 63–67, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Murugusundaramoorthy and H. M. Srivastava, “Neighborhoods of certain classes of analytic functions of complex order,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 5, no. 2, article 24, pp. 1–8, 2004. View at: Google Scholar | Zentralblatt MATH | MathSciNet - G. Murugusundaramoorthy and N. Magesh, “Starlike and convex functions of complex order involving the Dziok-Srivastava operator,”
*Integral Transforms and Special Functions*, vol. 18, no. 5-6, pp. 419–425, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. D. Bernardi, “Convex and starlike univalent functions,”
*Transactions of the American Mathematical Society*, vol. 135, pp. 429–446, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Murugusundaramoorthy and N. Magesh, “Linear operators associated with a subclass of uniformly $\beta $-convex functions,” in
*Proceedings of the National Conference on Geometry, Analysis, Fluid Mechanics and Computer Applications*, pp. 103–110, Kuvempu University, Karnataka, India, December 2004. View at: Google Scholar - H. S. Al-Amiri, “On a subclass of close-to-convex functions with negative coefficients,”
*Mathematica*, vol. 31(54), no. 1, pp. 1–7, 1989. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Silverman, “Integral means for univalent functions with negative coefficients,”
*Houston Journal of Mathematics*, vol. 23, no. 1, pp. 169–174, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. E. Littlewood, “On inequalities in theory of functions,”
*Proceedings of the London Mathematical Society*, vol. 23, pp. 481–519, 1925. View at: Google Scholar - A. W. Goodman, “Univalent functions and nonanalytic curves,”
*Proceedings of the American Mathematical Society*, vol. 8, pp. 598–601, 1957. View at: Publisher Site | Google Scholar | MathSciNet - S. Ruscheweyh, “Neighborhoods of univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 81, no. 4, pp. 521–527, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2010 G. Murugusundaramoorthy and N. Magesh. 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.