Journal of Function Spaces

Volume 2017 (2017), Article ID 9010964, 7 pages

https://doi.org/10.1155/2017/9010964

## A Subclass of Analytic Functions Related to -Uniformly Convex and Starlike Functions

^{1}Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan^{2}Department of Mathematics & Statistics, Riphah International University, Islamabad, Pakistan^{3}School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Correspondence should be addressed to Saqib Hussain; moc.oohay@htam_biqas

Received 26 January 2017; Accepted 20 April 2017; Published 23 May 2017

Academic Editor: Maria Alessandra Ragusa

Copyright © 2017 Saqib Hussain 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.

#### Abstract

We investigate some subclasses of -uniformly convex and -uniformly starlike functions in open unit disc, which is generalization of class of convex and starlike functions. Some coefficient inequalities, a distortion theorem, the radii of close-to-convexity, and starlikeness and convexity for these classes of functions are studied. The behavior of these classes under a certain modified convolution operator is also discussed.

#### 1. Introduction

Let be the class of all analytic functions in open unit disc , normalized by and Thus, any has the following Maclaurin’s series:

A function is said to be univalent if it never takes same value twice. By we mean the subclass of which is composed of univalent functions. By and we mean the well-known subclasses of that are, respectively, starlike and convex.

In 1991, Goodman [1, 2] introduced the classes and of uniformly convex and uniformly starlike functions, respectively. A function is uniformly convex if maps every circular arc contained in with center onto a convex arc. The function is uniformly starlike if maps every circular arc contained in with center onto a starlike arc with respect to . A more useful representation of and was given in [3–6] as

In 1999, for , Kanas and Wisniowska [7] introduced the class and as

Observe that , and , .

For fixed , these classes have a nice geometrical representation; for detail see [7–9].

A lot of authors obtain very useful properties of and and their generalization in several direction; for example, see [1, 2, 7, 8, 10, 11] and reference cited therein.

For , in [4] (see also [12]), Ronning introduced the following two important subclasses and as

Recently in [13] El-Ashwah et al. introduced two important subclass and of -uniformly convex starlike functions aswhere and .

Let be defined bythen the modified Hadmard product of and is defined by

We denote by subclass of consisting of functions having all negative coefficients in their Maclaurin’s series expansions, so any has a series of the form:

Let be the class of functions given in (1) for which , . Note that [11].

In recent years, more and more researchers are interested in the above defined classes (see [9, 11, 14–22]).

In this paper, by taking inspiration from the above cited paper, we introduce some new subclasses of analytic functions and obtain some interesting results.

*Definition 1. *For , , , and , a function is in class if and only if

Also

It is worth mentioning that, for special values of parameters, these classes were extensively studied by many authors; here we mention few of them.(1) [21].(2) [21].(3) [11].(4) [23].(5) [4].(6) [13].

Throughout the paper , , , and , unless otherwise stated.

#### 2. Main Results

Theorem 2. *A function given by (1) is in class ifwhere*

*Proof. *It is sufficient to prove that inequality (9) holds. As we knowthen inequality (9) can be written as This is, where then we haveNowand alsoFrom (18) and (19), we have The last expression is bounded below by if which completes the proof.

In the next theorem, we prove that condition (11) is also necessary for function .

Theorem 3. *Let be given by (1) and in ; then if and only if*

*Proof. *From Theorem 2, we need only to show that satisfies inequality (22). If , then by definition, we have Since is function of form (1) with the argument property given in class and letting in the above inequality, we havefor , and (24) leads to require inequality The functionis extremal function.

Corollary 4. *Let given in (1) be in class . Then*

Inequality (27) is attained for the function given in (26).

Theorem 5. *Let the function given in (1) be in class Then for *

The results in (28) are attained for the function given in (26) for .

*Proof. *As we know from Theorem 3Assimilarly This completes the proof.

Theorem 6. *Let the function given in (1) be in class Then for *

*Proof. *For given by (1), we haveIn view of Theorem 3,or, equivalently,A substitution from (35) into (33) yields inequality (32), which is required.

Theorem 7. *Let with argument property as in class . Define andwhere , .**Then function is in class if and only if it can be expressed aswhere and .*

*Proof. *Assume that Then it follows thatby Theorem 3, . Conversely, assume that the function defined by (1) belongs to class , and then Set and , . Then and this completes the proof.

Theorem 8. *Let . Then is close to convex of order in the disc , where*

*Proof. *As , where is close to convex of order , we haveas this expression is less than if By the fact that if and only if inequality (43) is true if or, equivalently,

Theorem 9. *Let . Then is close to convex of order in the , where*

*Proof. *As and is starlike of order , then we haveas The last expression is less than if Using the fact that if and only if (50) is true if Or equivalentlywhich is required.

Theorem 10. *Let . Then is convex of order in the , where*

*Proof. *Using the fact that is convex if and only if is starlike, following the lines of Theorem 9, we have the required results.

Theorem 11. *Let given by (6) be in class . Then , for*

*Proof. *We need to prove the largest such that From Theorem 3, we have By Cauchy-Schwarz inequality, we haveThus, it is sufficient to showFor Note thatWe need to showor equivalently is an increasing function for For in (65),which proves main assertion of Theorem 11.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

All authors jointly work on the results, and they read and approved the final manuscript.

#### Acknowledgments

The work here is supported by MOHE Grant: FRGS/1/2016/STG06/UKM/01/1.

#### References

- A. W. Goodman, “On uniformly convex functions,”
*Annales Polonici Mathematici*, vol. 56, no. 1, pp. 87–92, 1991. View at Google Scholar · View at 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 · View at Google Scholar · View at MathSciNet · View at Scopus - W. C. Ma and D. Minda, “Uniformly convex functions,”
*Annales Polonici Mathematici*, vol. 57, no. 2, pp. 165–175, 1992. View at Google Scholar · View at MathSciNet - F. R\o nning, “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 · View at Google Scholar · View at MathSciNet · View at Scopus - J. Sokol and A. Wisniowska-Wajnryb, “On some classes of starlike functions related with parabola,”
*Folia Sci. Univ. Tech. Resov.*, vol. 121, no. 18, pp. 35–42, 1993. View at Google Scholar · View at MathSciNet - J. Sokol and A. Wisniowska-Wajnryb, “On certain problem in the classes of
*k*-starlike functions,”*Computers & Mathematics with Applications*, vol. 62, no. 12, pp. 4733–4741, 2011. View at Publisher · View at Google Scholar · View at 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 · View at Google Scholar · View at MathSciNet - S. Kanas and H. M. Srivastava, “Linear operators associated with
*k*-uniformly convex functions,”*Integral Transforms and Special Function*, vol. 9, no. 2, pp. 121–132, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - A. Mannino, “Some inequalities concerning starlike and convex functions,”
*General Mathematics*, vol. 12, no. 1, pp. 5–12, 2004. View at Google Scholar · View at MathSciNet - S. Ponnusamy and M. Vuorinen, “Univalence and convexity properties for Gaussian hypergeometric functions,”
*The Rocky Mountain Journal of Mathematics*, vol. 31, no. 1, pp. 327–353, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - H. Silverman, “Univalent functions with negative coefficients,”
*Proceedings of the American Mathematical Society*, vol. 51, pp. 109–116, 1975. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Ronning, “Integral representation for bounded starlike functions,”
*Annales Polonici Mathematici*, vol. 60, no. 3, pp. 289–297, 1995. View at Google Scholar · View at MathSciNet - R. M. El-Ashwah, M. K. Aouf, A. A. Hassan, and A. H. Hassan, “Certain new classes of analytic functions with varying arguments,”
*Journal of Complex Analysis*, vol. 2013, Article ID 958210, 5 pages, 2013. View at Google Scholar · View at MathSciNet - R. M. Ali, S. R. Mondal, and V. Ravichandran, “On the Janowski convexity and starlikeness of the confluent hypergeometric function,”
*Bulletin of the Belgian Mathematical Society. Simon Stevin*, vol. 22, no. 2, pp. 227–250, 2015. View at Google Scholar · View at MathSciNet - R. M. Ali, V. Ravichandran, and N. Seenivasagan, “Subordination and superordination of the Liu-Srivastava linear operator on meromorphic functions,”
*Bulletin of the Malaysian Mathematical Sciences Society*, vol. 31, no. 2, pp. 193–207, 2008. View at Google Scholar · View at MathSciNet · View at Scopus - R. M. Ali and V. Ravichandran, “Uniformly convex and uniformly starlike functions,”
*Mathematics Newsletter*, vol. 21, pp. 16–30, 2011. View at Google Scholar - Ş. Altınkaya and S. Yalçın, “Coefficient estimates for two new subclasses of bi-univalent functions with respect to symmetric points,”
*Journal of Function Spaces*, Article ID 145242, 2014. View at Google Scholar · View at MathSciNet - M. K. Aouf, H. M. Hossen, and A. Y. Lashin, “On certain families of analytic functions with negative coefficients,”
*Indian Journal of Pure and Applied Mathematics*, vol. 31, no. 8, pp. 999–1015, 2000. View at Google Scholar · View at MathSciNet · View at Scopus - M. K. Aouf, A. A. Shamandy, A. O. Mostafa, and A. K. Wagdy, “Certain subclasses of uniformly starlike and convex functions defined by convolution with negative coefficients,”
*Matematichki Vesnik*, vol. 65, no. 1, pp. 14–28, 2013. View at Google Scholar · View at MathSciNet - A. Kamiński and S. Mincheva-Kaminska, “Compatibility conditions and the convolution of functions and generalized functions,”
*Journal of Function Spaces and Applications*, vol. 2013, Article ID 356724, 11 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - N. Magesh, “Certain subclasses of uniformly convex functions of order
*α*and type*β*with varying arguments,”*Journal of the Egyptian Mathematical Society*, vol. 21, no. 3, pp. 184–189, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - K. I. Noor, “Some properties of certain analytic functions,”
*Journal of Natural Geometry*, vol. 7, no. 1, pp. 11–20, 1995. View at Google Scholar · View at MathSciNet - R. Bharati, R. Parvatham, and A. Swaminathan, “On subclasses of uniformly convex functions and corresponding class of starlike functions,”
*Tamkang Journal of Mathematics*, vol. 28, no. 1, pp. 17–32, 1997. View at Google Scholar · View at MathSciNet