Abstract and Applied Analysis
Volume 2014 (2014), Article ID 703139, 6 pages
http://dx.doi.org/10.1155/2014/703139

Research Article

Convexity Properties for Certain Classes of Analytic Functions Associated with an Integral Operator

Ben Wongsaijai and Nattakorn Sukantamala

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 30 May 2014; Revised 4 September 2014; Accepted 4 September 2014; Published 14 October 2014

Academic Editor: Valery Serov

Copyright © 2014 Ben Wongsaijai and Nattakorn Sukantamala. 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 introduce a new form of generalized integral operator defined on the class of analytic functions . By making use of this novel integral operator, we give the convexity of other integral operators. We also briefly indicate the relevant connections of our presented results to the formerly reported results. Furthermore, other interesting properties are also discussed.

1. Introduction

In the field of geometric function theory, the class of univalent functions [1, 2] has been mainly studied. There are many distinguished geometric properties that played important role in the theory of univalent functions, such as starlikeness, convexity, and close-to-convexity (see, e.g., [3–5]). One of the generalizations of univalent functions is the theory of multivalent functions or -valent functions. Also, the geometric properties for the subclasses of -valent functions are investigated by many authors (see, e.g., [6–9]).

Let be an open unit disc in the complex plane. For a positive integer , denotes the class of -valent functions of the following form:

which is analytic in . In particular, we set . Furthermore, let be the class of analytic functions in of the following form:

A function is said to be -valently starlike of order in if satisfies

We denote this class by which was introduced by Patil and Thakare [10]. In particular, we set for a class of -valent starlike functions in . Denoted by , the subclass of consists of functions for which

On the other hand, a function is said to be -valently convex of order in if satisfies

We denote this class by which was introduced by Owa [11]. In particular, we set for a class of -valent convex functions in . Analogously, we denote that the subclass of consists of functions for which

By using the Alexander-type criterion, it follows that

The statement is also true if we replace and by and , respectively. Moreover, we note that

A function is said to be -uniformly -valent starlike of order in if satisfies

Furthermore, a function is said to be -uniformly -valent starlike of order in if satisfies

Both and are comprehensive classes of analytic functions that include various classes of analytic univalent functions as well as some very well-known ones. For example, in the case , we have and which are introduced by Bharati et al. [12]. For , is the class of -uniformly convex function [13]. In the special case, and , the class of uniformly starlike functions and of uniformly convex functions were introduced by Goodman [14, 15]. Using the Alexander type relation, statement (7) holds for the and ; that is,

Many researchers have studied the geometric properties of integral operators. The common investigation is finding sufficient conditions of integral operators in order to transform analytic functions into classes with each of those mentioned properties. The well-known integral transformation defining a subclass of univalent functions was introduced by Alexander in [16]. It is of the following form:

In [17], Kim and Merkes extended the integral operator (12) by introducing a complex parameter as

Another object of investigation for the studies of the integral operator by Pfaltzgraff [18] is defined by

Until now, the various generalized form of the integral operators in (13) and in (14) has been investigated. However, Breaz and Stanciu [19] introduced and studied the more general form of integral operator , which is

By setting appropriate values for the parameters , , and , integral operators that have been previously introduced can be obtained. In particular, if , then the integral operator becomes the integral operator introduced by D. Breaz and N. Breaz [20]. Also, when , the integral operator is exactly the integral operator defined by Breaz et al. [21]. The specialized form of and involving the Bessel functions was introduced and studied in [22–24]. In addition, the specific case for in (15), was investigated by Pescar in [25]. The univalence and their properties of the integral operators are reported in [26–30].

In [31], Bulut developed the integral operator which extends the class of analytic functions to the class of -valent functions ; that is,

By setting and , we obtain the integral operators and , respectively, which were introduced by Frasin [32]. Also, some properties of these integral operators have been studied in [32–34].

Recently, many authors modified integral operators associated with differential operator such as Salagean operator [35], Ruscheweyh operator [36], Al-Oboudi operator [37], and Carlson-Shaffer operator [38]. In [39], Frasin investigated one of the generalized integral operators by using the Hadamard product to demonstrate most of the previously defined integral operators. Frasin [39] defined the integral operator by

where , and the Hadamard product is defined by

where and . It was reported that, for appropriate functions , the integral operator generalizes many integral operators introduced and studied by several authors [40–44]. Moreover, the integral operator is generalized integral operators of those in (12)–(15). In a similar idea, can be extended to more generalized on to by

where , , and , and the Hadamard product is defined by

Certainly, the integral operator generalizes many operators when we choose suitable functions (see, e.g., [45–47]).

In fact, we can write the Hadamard product in the form where is analytic in and . Indeed,

which usually appears in most of integral operators and always belongs to the class . That is why we are interested in replacing the term in the integral operator (19) with general function in . Additionally, replacing the term with a function yields the integral operator, which is still contained in .

We now define the following general integral operator , for , , and by

where for all .

The main purpose of the paper is to investigate the sufficient conditions on convexity of the integral operator on classes , , and of analytic functions. Our main results will be applied to reinstate the results of former researches with related integral operators.

2. Main Results

In this section, we investigate sufficient conditions for the convexity of the integral operator which is defined by (22). For the convenience, we introduce the transformation operator by

where and is a nonnegative integer. In particular, we set .

We now prove a general property which guarantees the convexity of the proposed integral operator on the class .

Theorem 1. Let , , , and for . If satisfies

then the integral operator defined by (22) is -valently convex of order .

Proof. From the definition of integral opeartor in (22), we observe that . By calculating the first derivative of , we obtain

Differentiating on both sides of (25) logarithmically and multiplying by give

Since , it follows that

Also, since , we have

From (27) and (28), by taking the real part of (26), we obtain

Therefore, is -valently convex of order .

We note that by suitable functions , , and in Theorem 1, we obtain the earlier result. For example, if and , we obtain Theorem  2.1 in [44] and Theorems 2.1 and 2.2 in [48]. If and , by using the Alexander type relation (7), Theorem  1 in [19] is obtained.

Using the same method and technique as that in Theorem 1 with the nonnegativity of modulus of complex numbers, we are led easily to Theorem 2. The proof is omitted.

Theorem 2. Let and for . If satisfies

then the integral operator defined by (22) is -valently convex of order .

Theorem 2 generalizes many results proposed by several authors. For and , we obtain Theorem  2.1 in [26] and Theorem  2 in [49]. For and with the Alexander type relation (11), we obtain Theorem  2 in [46] and Theorem  2.1 in [31]. Also, Theorems 2.1 and 3.1 in [33] are obtained.

The following is a result on the transformation property of on the class .

Theorem 3. Let and for . If satisfies

then the integral operator defined by (22) is in the class . Furthermore, is -valently convex of order .

Proof. From (26), we obtain

Since , it follows that

Also, since , we have

By substitution of (33) and (34) into (32), we get

Therefore, . It follows that is -valently convex of order .

Remark 4. The parameter in Theorem 3 can be extended to the complex number and assumption (31) becomes

That is, Theorem 3 can be applied to the integral operator in case .

Over the past few decades, there are many studies on the sufficient conditions that make the integral operators univalent. In fact, the class of convex functions is a subclass of the class of all univalent functions in . Thus, it is interesting to observe that many results on the univalence property of integral operators follow the convexity property according to main results, especially Theorem 3 or Remark 4.

We now consider the integral operator defined in (17). In order to obtain the convexity of the integral operator by Theorem 3 or Remark 4, we set , , , , and

where . Other than that, the univalent property of is also obtained. This implies Theorem 3.1 of Frasin in [39]. Moreover, Frasin [39] noticed that for suitable functions , the integral operator generalizes many operators introduced by several authors, for instance, Theorem  1 in [20], Theorem  2.1 in [41], Theorem  2.3 in [42], and Theorem  2.3 in [43]. It is noteworthy to say that, under same assumptions, the former researches obtain only the univalence, while we obtain the stronger result, which is the convexity.

Our results can be used to explain the convexity of the other integral operators that are related to the Hadamad product as described next.

Remark 5. For , we set and take

where . Then, we can apply the main results to discuss the convexity of the integral operator defined by (19).

Remark 6. The main results are also applicable to the integral operator of the following form:

where . In order to apply the main results, by the Alexander-type criterion, we note that

The above statement also holds for the pairs of classes and .

Remark 7. For suitable functions , by Remarks 5 and 6, we obtain new results for the convexity of other integral operators, for example, and in [45], and in [46], and in [50], and so forth.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that improved the presentation of the paper. This research was supported by Science Achievement Scholarship of Thailand (SAST) and Department of Mathematics, Faculty of Science, Chiang Mai University.

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