Journal of Function Spaces
Volume 2017 (2017), Article ID 9010964, 7 pages
https://doi.org/10.1155/2017/9010964

Research Article

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

Saqib Hussain,¹ Akhter Rasheed,² Muhammad Asad Zaighum,² and Maslina Darus³

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

Correspondence should be addressed to Saqib Hussain; saqib_math@yahoo.com

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 as

where and .

Let be defined by

then 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 if

where

Proof. It is sufficient to prove that inequality (9) holds. As we know

then inequality (9) can be written as

This is,

where

then we have

Now

and also

From (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 have

for , and (24) leads to require inequality

The function

is 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 3

As

similarly

This completes the proof.

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

Proof. For given by (1), we have

In 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 and

where , .
Then function is in class if and only if it can be expressed as

where and .

Proof. Assume that

Then it follows that

by 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 have

as

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 have

as

The last expression is less than if

Using the fact that if and only if

(50) is true if

Or equivalently

which 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 have

Thus, it is sufficient to show

For

Note that

We need to show

or 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.

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