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Some Applications of Second-Order Differential Subordination on a Class of Analytic Functions Defined by Komatu Integral Operator  [PDF]
Serap Bulut
ISRN Mathematical Analysis , 2014, DOI: 10.1155/2014/606235
Abstract: We introduce a new class of analytic functions by using Komatu integral operator and obtain some subordination results. 1. Introduction, Definitions, and Preliminaries Let be the set of real numbers, the set of complex numbers, be the set of positive integers, and Let be the class of analytic functions in the open unit disk and the subclass of consisting of the functions of the form Let be the class of all functions of the form which are analytic in the open unit disk with Also let denote the subclass of consisting of functions which are univalent in . A function analytic in is said to be convex if it is univalent and is convex. Let denote the class of normalized convex functions in . If and are analytic in , then we say that is subordinate to , written symbolically as if there exists a Schwarz function which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence [1, page 4]: Let be a function and let be univalent in . If is analytic in and satisfies the (second-order) differential subordination then is called a solution of the differential subordination. The univalent function is called a dominant of the solutions of the differential subordination, or more simply a dominant, if for all satisfying (13). A dominant , which satisfies for all dominants of (13), is said to be the best dominant of (13). Recently, Komatu [2] introduced a certain integral operator defined by Thus, if is of the form (5), then it is easily seen from (14) that (see [2]) Using the relation (15), it is easy verify that We note the following.(i)For and ( is any integer), the multiplier transformation was studied by Flett [3] and S?l?gean [4].(ii)For and ( ), the differential operator was studied by S?l?gean [4].(iii)For and ( is any integer), the operator was studied by Uralegaddi and Somanatha [5].(iv)For , the multiplier transformation was studied by Jung et al. [6]. Using the operator , we now introduce the following class. Definition 1. Let be the class of functions satisfying where , , and is the Komatu integral operator. In order to prove our main results, we will make use of the following lemmas. Lemma 2 (see [7]). Let be a convex function with and let be a complex number with . If and then where The function is convex and is the best dominant. Lemma 3 (see [8]). Let , , and let Let be an analytic function in with and suppose that If is analytic in and then where is a solution of the differential equation given by Moreover is the best dominant. 2. Main Results Theorem 4. The set is convex.
Some differential subordination and superordination properties of symmetric analytic functions involving Noor integral operator  [cached]
Ali Muhammad,Amjad Ullah Khattak
Le Matematiche , 2012,
Abstract: In this paper, we obtain some interesting properties of differential subordination and superordination for the classes of symmetric functions analytic in the unit disc, by applying Noor integral operator. We investigate several sandwich theorems on the basis of this theory.
On Subordination Result Associated with Certain Subclass of Analytic Functions Involving Salagean Operator
Sevtap Sümer Eker,Bilal ?eker,Shigeyoshi Owa
Journal of Inequalities and Applications , 2007, DOI: 10.1155/2007/48294
Abstract: We obtain an interesting subordination relation for Salagean-type certain analytic functions by using subordination theorem.
On Subordination Result Associated with Certain Subclass of Analytic Functions Involving Salagean Operator  [cached]
Eker Sevtap Sümer,?eker Bilal,Owa Shigeyoshi
Journal of Inequalities and Applications , 2007,
Abstract: We obtain an interesting subordination relation for Salagean-type certain analytic functions by using subordination theorem.
On a Class of Analytic Functions Defined by an Integral Operator  [PDF]
Pravati Sahoo,Saumya Singh
Journal of Mathematics , 2013, DOI: 10.1155/2013/923167
Abstract: We define a new subclass by using an integral operator . We find a coefficient inequality and using that we derive many sharp results. These results generalize many results which are existing in the literature. 1. Introduction and Preliminaries Let denote the class of functions of the form which are analytic in the unit disc . For and for , , the integral operator is defined by where is the Pochhammer symbol given by For , , is defined by Komatu in [1, 2]. Here, The operator is the Bernardi operator [3, 4]. In fact the operator is related rather closely to the Beta or Euler transformation. Moreover, for , the operator was used by Owa and Srivastava [5–8]. For , , we define a class of all analytic functions involving the integral operator, , by The aim of this paper is to study the class and find the similar type of results proved by Frasin in [9], where the author has defined similar type of class involving the operator Also in [10], the authors studied the similar type class involving the well-known Salagean operator. 2. Definitions and Lemmas Definition 1. Let be analytic and univalent in . If is analytic in , , and , then we say that the function is subordinate to in , and we write . Definition 2 (subordinating factor sequence). A sequence of complex numbers is called a subordinating sequence if, whenever is analytic, univalent, and convex in , we have the subordination given by Lemma 3 (see [11]). A sequence is a subordinating factor sequence if and only if Lemma 4. If where , and then . Proof. It is sufficient to show that Now, we have The above expansion is bounded by if hence the proof follows from (10). Let denote the class of functions whose coefficients satisfy the condition (10). So . 3. Main Results By using the technique used earlier by Attiya [12] and Singh [13], we state and prove the following theorem. Theorem 5. Let the function defined by (1) be in the class , where . Also let denote the class of functions which are convex and univalent in . Then, The constant is the best estimate. Proof. and let Then, Thus by the Definition 2 and (14) will hold if the sequence is a subordinating factor sequence, with , in view of Lemma 3, this will be the case if and only if Now Case I ( ). From (19), we obtain Since is an increasing function of , so Case II ( ). From (19), we obtain Since is decreasing function of , so Thus, (18) holds true in . This prove the inequality (14). The inequality (15) follows by taking the convex function in (14). To prove the sharpness of the constant, we suppose that the function given by from (14), we have After a
New Classes of Analytic Functions Involving Generalized Noor Integral Operator
Ibrahim RabhaW,Darus Maslina
Journal of Inequalities and Applications , 2008,
Abstract: The present article investigates new classes of functions involving generalized Noor integral operator. Some properties of these functions are studied including characterization and distortion theorems. Moreover, we illustrate sufficient conditions for subordination and superordination for analytic functions.
New Classes of Analytic Functions Involving Generalized Noor Integral Operator  [cached]
Rabha W. Ibrahim,Maslina Darus
Journal of Inequalities and Applications , 2008, DOI: 10.1155/2008/390435
Abstract: The present article investigates new classes of functions involving generalized Noor integral operator. Some properties of these functions are studied including characterization and distortion theorems. Moreover, we illustrate sufficient conditions for subordination and superordination for analytic functions.
Strong Differential Subordinations Obtained with New Integral Operator Defined by Polylogarithm Function  [PDF]
K. AL-Shaqsi
International Journal of Mathematics and Mathematical Sciences , 2014, DOI: 10.1155/2014/260198
Abstract: By using the polylogarithm function, a new integral operator is introduced. Strong differential subordination and superordination properties are determined for some families of univalent functions in the open unit disk which are associated with new integral operator by investigating appropriate classes of admissible functions. New strong differential sandwich-type results are also obtained. 1. Introduction Let denote the class of analytic function in the open unit disk . For a positive integer and , let and let . We also denote by the subclass of , with the usual normalization . Let and be formal Maclaurin series. Then, the Hadamard product or convolution of and is defined by the power series . Let the functions and in ; then we say that is subordinate to in , and write , if there exists a Schwarz function in with and such that in . Furthermore, if the function is univalent in , then and (cf [1–3]). Let denote the well-known generalization of the Riemann zeta and polylogarithm functions, or simply the th order polylogarithm function, given by where any term with is excluded; see Lerch [4] and also [5, Sections 1.10 and 1.12]. Using the definition of the Gamma function [5, page 27], a simply transformation produces the integral formula Note that is Koebe function. For more details about polylogarithms in theory of univalent functions, see Ponnusamy and Sabapathy [6] and Ponnusamy [7]. Now, for , we defined the following integral operator: where , and . We also note that the operator defined by (4) can be expressed by the series expansion as follows: Obviously, we have, for , Moreover, from (5), it follows that We note that,(i)for and ( is any integer), the multiplier transformation was studied by Flett [8] and S?l?gean [9];(ii)for and ( ), the differential operator was studied by S?l?gean [9];(iii)for and ( is any integer), the operator was studied by Uralegaddi and Somanatha [10];(iv)for , the multiplier transformation was studied by Jung et al. [11];(v)for , the integral operator was studied by Komatu [12].To prove our results, we need the following definition and theorems considered by Antonino and Romaguera [13], Antonino [14], G. I. Oros and G. Oros [15], and Oros [16]. Definition 1 (see [13] cf [14, 15]). Let be analytic in and let be analytic and univalent in . Then, the function is said to be strongly subordinate to , or is said to be strongly superordinate to , written as , if, for , as the function of is subordinate to . We note that if and only if and . Definition 2 ([15] cf [1]). Let and let be univalent in . If is analytic in and satisfies
Differential Subordination with Generalized Derivative Operator of Analytic Functions  [PDF]
Entisar El-Yagubi,Maslina Darus
Chinese Journal of Mathematics , 2014, DOI: 10.1155/2014/656258
Abstract: Motivated by generalized derivative operator defined by the authors (El-Yagubi and Darus, 2013) and the technique of differential subordination, several interesting properties of the operator are given. 1. Introduction Let denote the class of functions of the form which are analytic in the open unit disk . Also let be the the subclass of consisting of all functions which are univalent in . We denote by and the familiar subclasses of consisting of functions which are, respectively, starlike of order and convex of order in :. Let be the class of holomorphic function in unit disk . For and we let Let two functions given by and be analytic in . Then the Hadamard product (or convolution) of the two functions , is defined by Recall that the function is subordinate to if there exists the Schwarz function , analytic in , with and such that , . We denote this subordination by . If is univalent in , then the subordination is equivalent to and . Let and be univalent in . If is analytic in and satisfies the (second order) differential subordination then is called a solution of the differential subordination. The univalent function is called a dominant of the solutions of the differential subordination, or more simply a dominant, if for all satisfying (5). A dominant that satisfies for all dominants of (5) is said to be the best dominant of (5) (note that the best dominant is unique up to a rotation of ). In order to prove the original results we need the following lemmas. Lemma 1 (see [1]). Let be a convex function with and let be a complex number with . If and then where The function is convex and is the best dominant. Lemma 2 (see [2]). Let be a convex function in and let where and is a positive integer. If is analytic in and then and this result is sharp. Lemma 3 (see [3]). Let ; if then belongs to the class of convex functions. We now state the following generalized derivative operator [4]: where , for , and is the Pochhammer symbol defined by Here can also be written in terms of convolution as To prove our results, we need the following inclusion relation: where is analytic function given by . 2. Main Results In the present paper, we will use the method of differential subordination to derive certain properties of generalised derivative operator . Note that differential subordination has been studied by various authors, and here we follow similar works done by Oros [5] and G. Oros and G. I. Oros [6]. Definition 4. For , and , let denote the class of functions which satisfy the condition Also, let denote the class of functions which satisfy the condition Remark
Subordination and superordination for certain analytic functions containing fractional integral
Rabha W. Ibrahim,Maslina Darus,Shaher Momani
Surveys in Mathematics and its Applications , 2009,
Abstract: The purpose of the present article is to derive some subordination and superordination results for certain normalized analytic functions involving fractional integral operator. Moreover, this result is applied to find a relation between univalent solutions for fractional differential equation.
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