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On the Fekete-Szeg? Problem for a Class of Analytic Functions  [PDF]
Zhigang Peng
ISRN Mathematical Analysis , 2014, DOI: 10.1155/2014/861671
Abstract: Let denote the class of functions which are analytic in the unit disk and given by the power series . Let be the class of convex functions. In this paper, we give the upper bounds of for all real number and for any in the family , Re for??some . 1. Introduction Let denote the class of functions which are analytic in the unit disk and satisfy . The set of all functions that are univalent will be denoted by . Let and be the classes of convex, starlike of order and close-to-convex functions, respectively. Fekete and Szeg? [1] proved that holds for any and that this inequality is sharp. The coefficient functional on in plays an important role in function theory. For example, , where is the Schwarzian derivative. The problem of maximizing the absolute value of the functional is called the Fekete-Szeg? problem. In the literature, there exist a large number of results about the Fekete-Szeg? problem (see, for instance, [2–11]). For and , let denote the class of functions satisfying and for some . Al-Abbadi and Darus [7] investigated the Fekete-Szeg? problem on the class . Let be the class of functions in satisfying the inequality for some function . In [11], Srivastara et al. studied the Fekete-Szeg? problem on the class for by proving that Srivastara et al. held that the inequality (5) was sharp. However, the extremal function given in [11] did not exist in the case of . In this paper, we solve the Fekete-Szeg? problem for the family As a corollary of the main result, we find the sharp upper bounds for absolute value of the Fekete-Szeg? functional for the class defined by Clearly, is a subclass of . In the case of , we get sharp estimation of the absolute value of the Fekete-Szeg? functional for the class and for all real number , which prove that the inequality (5) is not sharp actually when . 2. Main Result Let be the class of functions that are analytic in and satisfy for all . The following two lemmas can be found in [2]. Lemma 1 (see [2]). If is in the class , then, for any complex number , one has . The inequality is sharp. Lemma 2 (see [2]). If is in the class and is a complex number, then . The inequality is sharp. Theorem 3. If is in the class and is a real number, then Proof. By definition, is in the class if and only if there exists a function such that . A simple computation shows . Thus, So, by Lemmas 1 and 2, we have Putting and , we get from (10) that , where Since and , we will calculate the maximum value of for . Case 1. Suppose . Then it follows from (11) that Since does not have a local maximum at any point of the open rectangle . Hence,
On the Fekete-Szeg? inequality for Certain class of analytic functions  [PDF]
S. Sivaprasad Kumar,Virendra Kumar
Mathematics , 2012,
Abstract: In the present investigation, we derive Fekete-Szeg\"{o} inequality for the class $\mathcal{S}^{\alpha}_{\mathscr{L}_{g}}(\phi)$, introduced here. In addition to that, certain applications of our results are also discussed.
Fekete-Szeg? Type Coefficient Inequalities for Certain Subclass of Analytic Functions and Their Applications Involving the Owa-Srivastava Fractional Operator  [PDF]
Serap Bulut
International Journal of Analysis , 2014, DOI: 10.1155/2014/490359
Abstract: A new subclass of analytic functions is introduced. For this class, firstly the Fekete-Szeg? type coefficient inequalities are derived. Various known or new special cases of our results are also pointed out. Secondly some applications of our main results involving the Owa-Srivastava fractional operator are considered. Thus, as one of these applications of our result, we obtain the Fekete-Szeg? type inequality for a class of normalized analytic functions, which is defined here by means of the Hadamard product (or convolution) and the Owa-Srivastava fractional operator. 1. Introduction and Definitions Let denote the class of functions of the form which are analytic in the unit disk Also let denote the subclass of consisting of univalent functions in . Fekete and Szeg? [1] proved a noticeable result that the estimate holds for . The result is sharp in the sense that for each there is a function in the class under consideration for which equality holds. The coefficient functional on represents various geometric quantities as well as in the sense that this behaves well with respect to the rotation; namely, In fact, rather than the simplest case when we have several important ones. For example, represents , where denotes the Schwarzian derivative Moreover, the first two nontrivial coefficients of the th root transform of with the power series (1) are written by so that where Thus it is quite natural to ask about inequalities for corresponding to subclasses of . This is called Fekete-Szeg? problem. Actually, many authors have considered this problem for typical classes of univalent functions (see, e.g., [1–12]). For two functions and , analytic in?? , we say that the function is subordinate to in , and we write if there exists a Schwarz function , analytic in , with such that In particular, if the function is univalent in , the above subordination is equivalent to Let be an analytic function with which maps the open unit disk onto a star-like region with respect to and is symmetric with respect to the real axis. This paper contains analogues of (3) for the following classes of analytic functions. Definition 1. Let A function is said to be in the class if it satisfies the following subordination condition: where is defined to be the same as above for . Remark 2. (i) If we set in Definition 1, then we have the class which consists of functions satisfying This class was introduced by Bansal [13].(ii)If we set in Definition 1, then we have a new class which consists of functions satisfying Taking in (25), we have the class which consists of functions satisfying
On the Fekete-Szeg? problem for concave univalent functions  [PDF]
B. Bhowmik,S. Ponnusamy,K-J. Wirths
Mathematics , 2010,
Abstract: We consider the Fekete-Szeg\"o problem with real parameter $\lambda$ for the class $Co(\alpha)$ of concave univalent functions.
Fekete-Szeg? problem and Second Hankel Determinant for a class of bi-univalent functions  [PDF]
N. Magesh,J. Yamini
Mathematics , 2015,
Abstract: In this sequel to the recent work (see Azizi et al., 2015), we investigate a subclass of analytic and bi-univalent functions in the open unit disk. We obtain bounds for initial coefficients, the Fekete-Szeg\"o inequality and the second Hankel determinant inequality for functions belonging to this subclass. We also discuss some new and known special cases, which can be deduced from our results.
Coefficient determinants involving many Fekete-Szeg$\ddot{O}$-type parameters  [PDF]
K. O. Babalola
Mathematics , 2013,
Abstract: We extend our definition (in a recent paper \cite{KB}) of the coefficient determinants of analytic mappings of the unit disk to include many Fekete-Szeg$\ddot{o}$-type parameters, and compute the best possible bounds on certain specific determinants for the choice class of starlike functions.
一类λ-对数Bazilevic函数的Fekete-Szeg?不等式
FEKETE-SZEG? INEQUALITY FOR A CLASS OF λ-LOGARITHMIC BAZILEVIC FUNCTIONS
 [PDF]

作者,鲍春梅,李书海,马丽娜
- , 2017,
Abstract: 本文研究了一类λ-对数Bazilevic函数的Fekete-Szeg?不等式.利用分类讨论的方法获得了|a3-μa22|的精确估计,推广了一些已有的相关结果.
In this paper, we discuss the Fekete-Szeg? inequality of a class of λ-logarithmic Bazilevic function. Using the methods of the classiflcation, we obtain the accurate estimation of|a3-μa22|, which generalizes some known results
Fekete-Szeg? Inequalities for Starlike Functions with Respect to -Symmetric Points of Complex Order  [PDF]
M. K. Aouf,R. M. El-Ashwah,S. M. El-Deeb
Journal of Complex Analysis , 2014, DOI: 10.1155/2014/131475
Abstract: Sharp upper bounds of for the function belonging to certain subclass of starlike functions with respect to -symmetric points of complex order are obtained. Also, applications of our results to certain functions defined through convolution with a normalized analytic function are given. In particular, Fekete-Szeg? inequalities for certain classes of functions defined through fractional derivatives are obtained. 1. Introduction Let denote the class of analytic functions of the following form: And let be the subclass of , which are univalent functions. Let be given by the following: The Hadamard product (or convolution) of and is given by If and are analytic functions in , we say that is subordinate to , written if there exists a Schwarz function , which is analytic in with and for all , such that . Furthermore, if the function is univalent in , then we have the following equivalence (see [1, 2]): Sakaguchi [3] introduced a class of functions starlike with respect to symmetric points, which consists of functions satisfying the inequality Chand and Singh [4] introduced a class of functions starlike with respect to -symmetric points, which consists of functions satisfying the inequality where Al-Shaqsi and Darus [5] defined the linear operator as follows: and in general where In this paper, we define the following class ( ) as follows. Definition 1. Let be univalent starlike function with respect to which maps the unit disk onto a region in the right half plane which is symmetric with respect to the real axis. Let be a complex number and let . Then functions are in the class if where is defined by (9) and is defined by (7). We note that for suitable choices of , , , , and we obtain the following subclasses:(i) (see Al-Shaqsi and Darus [6]),(ii) (see Al-Shaqsi and Darus [6]),(iii) (see Al-Shaqsi and Darus [6]),(iv) (see Shanmugam et al. [7]),(v) (see Sakaguchi [3]),(vi) (see Shanthi et al. [8] and Al-Shaqsi and Darus [9]),(vii) (see Ma and Minda [10]),(viii) (see Janowski [11]),(ix) and (see Ravichandran et al. [12]),(x) (see Nasr and Aouf [13]),(xi) (see Nasr and Aouf [14] and Aouf et al. [15]),(xii) (see Libera [16]),(xiii) (see Chichra [17]),(xiv) and (see Aouf and Silverman [18]),(xv) (see Keogh and Markes [19]).Also, we note the following: In this paper, we obtain the Fekete-Szeg? inequalities for the functions in the class . We also give application of our results to certain functions defined through convolution and, in particular, we consider the class defined by fractional derivatives. 2. Fekete-Szeg? Problem To prove our results, we need the following
Coefficients bounds in some subclass of analytic functions
Janusz Sokol,Deepak Bansal
Tamkang Journal of Mathematics , 2012, DOI: 10.5556/j.tkjm.43.2012.621-630
Abstract: In this paper we consider a class of analytic functions introduced by Mishra and Gochhayat, {it Fekete-Szeg"o problem for a class defined by an integral operator}, Kodai Math. J., 33(2010) 310--328, which is connected with $k$-starlike functions through Noor operator. We find inclusion relations and coefficients bounds in this class.
The Fekete-Szeg\"lcurlyborcurlyb Problem for -Valently Janowski Starlike and Convex Functions  [PDF]
Toshio Hayami,Shigeyoshi Owa
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/583972
Abstract: For -valently Janowski starlike and convex functions defined by applying subordination for the generalized Janowski function, the sharp upper bounds of a functional related to the Fekete-Szeg? problem are given. 1. Introduction Let denote the family of functions normalized by which are analytic in the open unit disk . Furtheremore, let be the class of functions of the form which are analytic and satisfy in . Then, a function is called the Schwarz function. If satisfies the following condition for some complex number , then is said to be -valently starlike function of complex order . We denote by the subclass of consisting of all functions which are -valently starlike functions of complex order . Similarly, we say that is a member of the class of -valently convex functions of complex order in if satisfies for some complex number . Next, let and . Then, the condition of the definition of is equivalent to We denote by the distance between the boundary line of the half plane satisfying the condition (1.5) and the point . A simple computation gives us that that is, that is always equal to regardless of . Thus, if we consider the circle with center at and radius , then we can know the definition of means that is covered by the half plane separated by a tangent line of and containing . For , the same things are discussed by Hayami and Owa [3]. Then, we introduce the following function: which has been investigated by Janowski [4]. Therefore, the function given by (1.7) is said to be the Janowski function. Furthermore, as a generalization of the Janowski function, Kuroki et al. [6] have investigated the Janowski function for some complex parameters and which satisfy one of the following conditions: Here, we note that the Janowski function generalized by the conditions (1.8) is analytic and univalent in , and satisfies . Moreover, Kuroki and Owa [5] discussed the fact that the condition can be omitted from among the conditions in (1.8)-(i) as the conditions for and to satisfy . In the present paper, we consider the more general Janowski function as follows: for some complex parameter and some real parameter . Then, we don't need to discuss the other cases because for the function: letting and replacing by in (1.10), we see that maps onto the same circular domain as . Remark 1.1. For the case in (1.9), we know that maps onto the following half plane: and for the case in (1.9), maps onto the circular domain Let and be analytic in . Then, we say that the function is subordinate to in , written by if there exists a function such that . In particular, if is univalent
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