Abstract and Applied Analysis

Volume 2014 (2014), Article ID 703139, 6 pages

http://dx.doi.org/10.1155/2014/703139

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

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.

#### References

- C. Pommerenke,
*Univalent Functions*, Vandemhoeck and Ruprecht, Göttingen, Germany, 1975. View at MathSciNet - A. W. Goodman,
*Univalent Functions*, vol. 1-2, Mariner, Tampa, Fla, USA, 1983. - M. I. S. Robertson, “On the theory of univalent functions,”
*Annals of Mathematics*, vol. 37, no. 2, pp. 374–408, 1936. View at Publisher · View at Google Scholar · View at MathSciNet - A. Schild, “On a class of univalent, star shaped mappings,”
*Proceedings of the American Mathematical Society*, vol. 9, pp. 751–757, 1958. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. P. Merkes, “On products of starlike functions,”
*Proceedings of the American Mathematical Society*, vol. 13, pp. 479–492, 2013. View at Google Scholar - M. Nunokawa, “On the multivalent functions,”
*Indian Journal of Pure and Applied Mathematics*, vol. 20, no. 6, pp. 577–582, 1989. View at Google Scholar · View at MathSciNet - M. Nunokawa and S. Owa, “On certain subclass of analytic functions,”
*Indian Journal of Pure and Applied Mathematics*, vol. 19, no. 1, pp. 51–54, 1988. View at Google Scholar · View at MathSciNet - J. Dziok, “Applications of the Jack lemma,”
*Acta Mathematica Hungarica*, vol. 105, no. 1-2, pp. 93–102, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. E. Livingston, “
*p*-valent close-to-convex functions,”*Soochow Journal of Mathematics*, vol. 21, pp. 161–179, 1995. View at Google Scholar - D. A. Patil and N. K. Thakare, “On convex hulls and extreme points of
*p*-valent starlike and convex classws with applications,”*Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie*, vol. 27, no. 75, pp. 145–160, 1983. View at Google Scholar - S. Owa, “On certain classes of $p$-valent functions with negative coefficients,”
*Simon Stevin A: Quarterly Journal of Pure and Applied Mathematics*, vol. 59, no. 4, pp. 385–402, 1985. 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 Zentralblatt MATH · 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 · View at Scopus - A. W. Goodman, “On uniformly starlike functions,”
*Journal of Mathematical Analysis and Applications*, vol. 155, pp. 364–370, 1991. View at Publisher · View at Google Scholar - A. W. Goodman, “On uniformly convex functions,”
*Polska Akademia Nauk: Annales Polonici Mathematici*, vol. 56, no. 1, pp. 87–92, 1991. View at Google Scholar · View at MathSciNet - J. W. Alexander, “Functions which map the interior of the circle upon simple regions,”
*Annals of Mathematics*, vol. 14, pp. 12–22, 1915. View at Google Scholar - Y. J. Kim and E. P. Merkes, “On an integral of powers of a spirallike function,”
*Kyungpook Mathematical Journal*, vol. 12, pp. 249–252, 1972. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. A. Pfaltzgraff, “Univalence of the integral of
*f*′(*z*)^{λ},”*The Bulletin of the London Mathematical Society*, vol. 7, no. 3, pp. 254–256, 1975. View at Publisher · View at Google Scholar · View at MathSciNet - D. Breaz and L. Stanciu, “Some properties of a general integral operator,”
*Bulletin of the Transilvania University of Brasov Series III: Mathematics, Informatics, Physics*, vol. 5, no. 54, pp. 67–72, 2012. View at Google Scholar · View at MathSciNet - D. Breaz and N. Breaz, “Two integral operator,”
*Studia Universitatis Babes-Bolyai*, vol. 47, no. 3, pp. 13–19, 2002. View at Google Scholar - D. Breaz, S. Owa, and N. Breaz, “A new integral univalent operator,”
*Acta Universitatis Apulensis. Mathematics-Informatics*, no. 16, pp. 11–16, 2008. View at Google Scholar · View at MathSciNet - E. Deniz, H. Orhan, and H. M. Srivastava, “Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions,”
*Taiwanese Journal of Mathematics*, vol. 15, no. 2, pp. 883–917, 2011. View at Google Scholar · View at MathSciNet · View at Scopus - E. Deniz, “Convexity of integral operators involving generalized Bessel functions,”
*Integral Transforms and Special Functions*, vol. 24, no. 3, pp. 201–216, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. A. Frasin, “Sufficient conditions for integral operator defined by Bessel functions,”
*Journal of Mathematical Inequalities*, vol. 4, no. 2, pp. 301–306, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Pescar, “The univalence and the convexity properties for a new integral operator,”
*Studia Universitatis Babes-Bolyai*, vol. 56, no. 4, pp. 65–69, 2011. View at Google Scholar · View at MathSciNet - D. Breaz, “A convexity property for an integral operator on the class
*𝒮*_{p}(*β*),”*Journal of Inequalities and Applications*, vol. 2008, Article ID 143869, 4 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - B. A. Frasin, “Some sufficient conditions for certain integral operator,”
*Journal of Inequalities and Applications*, vol. 2, no. 4, pp. 527–535, 2008. View at Google Scholar - S. Bulut, “A note on the paper of Breaz and Guney,”
*Journal of Mathematical Inequalities*, vol. 2, no. 4, pp. 549–553, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - D. Breaz and V. Pescar, “Univalence conditions for some general integral operators,”
*Banach Journal of Mathematical Analysis*, vol. 2, no. 1, pp. 53–58, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Breaz and H. O. Guney, “The integral operator on the classes ${\mathcal{S}}_{\alpha}^{*}$(
*b*) and ${\mathcal{C}}_{\mathit{\alpha}}$(*b*),”*Journal of Mathematical Inequalities*, vol. 2, no. 1, pp. 97–100, 2008. View at Google Scholar - S. Bulut, “Convexity properties of a new general integral operator of
*p*-valent functions,”*Mathematical Journal of Okayama University*, vol. 56, pp. 171–178, 2014. View at Google Scholar · View at MathSciNet - B. A. Frasin, “New general integral operators of $p$-valent functions,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 10, no. 4, article 109, 2009. View at Google Scholar · View at MathSciNet - B. A. Frasin, “Convexity of integral operators of $p$-valent functions,”
*Mathematical and Computer Modelling*, vol. 51, no. 5-6, pp. 601–605, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Mohammed, M. Darus, and D. Breaz, “New criterion for starlike integral operators,”
*Analysis in Theory and Applications*, vol. 29, no. 1, pp. 21–26, 2013. View at Google Scholar · View at MathSciNet - G. S. Salagean, “Subclasses of univalent functions,” in
*Complex Analysis*, vol. 1013 of*Lecture Notes in Mathematics*, pp. 362–372, Springer, Berlin, Germany, 1983. View at Google Scholar - S. Ruscheweyh, “New criteria for univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 49, no. 1, pp. 109–115, 1975. View at Publisher · View at Google Scholar · View at MathSciNet - F. M. Al-Oboudi, “On univalent functions defined by a generalized Sălăgean operator,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2004, no. 27, pp. 1429–1436, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. A. Frasin, “General integral operator defined by Hadamard product,”
*Matematichki Vesnik*, vol. 62, no. 2, pp. 127–136, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - D. Breaz, H. Guney, and G. Salagean, “A new integral operator,” in
*Proceedings of the 7th Joint Conference on Mathematics and Computer Science*, Cluj, Romania, July 2008. - C. Selvaraj and K. R. Karthikeyan, “Sufficient conditions for univalence of a general integral operator,”
*Bulletin of the Korean Mathematical Society*, vol. 46, no. 2, pp. 367–372, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - G. I. Oros and D. Breaz, “Sufficient conditions for univalence of an integral operator,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 127645, 7 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Bulut, “Sufficient conditions for univalence of an integral operator defined by Al-Oboudi differential operator,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 957042, 5 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Bulut, “Some properties for an integral operator defined by Al-Oboudi differential operator,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 9, no. 4, article 115, 2008. View at Google Scholar · View at MathSciNet - R. M. El-Ashwah, M. K. Aouf, A. Shamandy, and S. M. El-Deeb, “Properties of certain classes of
*p*-valent functions associated with new integral operators,”*The American Journal of Pure and Applied Mathematics*, vol. 2, pp. 79–80, 2013. View at Google Scholar - G. Saltik Ayhanoz and E. Kadioglu, “Some results of
*p*-valent functions defined by integral operators,”*Acta Universitatis Apulensis. Mathematics. Informatics*, no. 32, pp. 69–85, 2012. View at Google Scholar · View at MathSciNet - G. Saltik, E. Deniz, and E. Kadioglu, “Two new general $p$-valent integral operators,”
*Mathematical and Computer Modelling*, vol. 52, no. 9-10, pp. 1605–1609, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. Breaz and N. Breaz, “Some convexity properties for a general integral operator,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 7, no. 5, article 177, 8 pages, 2006. View at Google Scholar · View at MathSciNet - A. Mohammed, M. Darus, and D. Breaz, “Some properties for certain integral operators,”
*Acta Universitatis Apulensis—Matematics—Informatics*, no. 23, pp. 79–89, 2010. View at Google Scholar · View at MathSciNet - E. Deniz, M. Caglar, and H. Orhan, “Some convexity properties for two new $p$-valent integral operators,”
*Hacettepe Journal of Mathematics and Statistics*, vol. 40, no. 6, pp. 829–837, 2011. View at Google Scholar · View at MathSciNet · View at Scopus