A Subclass of Meromorphic Close-to-Convex Functions of Janowski's Type
We introduce a subclass of functions which are analytic in the punctured unit disk and meromorphically close-to-convex. We obtain some coefficients bounds and some argument and convolution properties belonging to this class.
Let be the set of all analytic functions on the open unit disk , and let be the subclass of which contains the functions normalized by and . A function is said to be starlike of order if and only if for some and for all . The class of starlike functions of order is denoted by .
If and are analytic in , we say that is subordinate to , written as follows: if there exists a Schwarz function , which is analytic in with such that In particular, if the function is univalent in , the above subordination is equivalent to The Hadamard product (or convolution) of two series is defined by For a convex function , it follows from Alexander’s Theorem that is a starlike function. In view of the identity , it is then clear that the classes of convex and starlike functions can be unified by considering functions satisfying is starlike for a fixed function .
Though the convolution of two univalent (or starlike) functions does not need be univalent, it is well-known that the classes of starlike, convex, and close-to-convex functions are closed under convolution with convex functions. These results were later extended to convolution with prestarlike functions. For , the class of prestarlike functions of order is defined by while consists of satisfying .
By using the convex hull method [1, 2] and the method of differential subordination , Shanmugam  introduced and investigated convolution properties of various subclasses of analytic functions. Ali et al.  and Supramaniam et al.  investigated these properties for subclasses of multivalent starlike and convex functions. And Chandrashekar et al.  also investigated these properties for the functions with respect to symmetric points, conjugate, or symmetric conjugate points. More results using the convex hull method and the method of differential subordination can be found in [8, 9].
Let denote the class of all univalent meromorphic functions normalized by which are analytic in the punctured unit disk We denote by the subclass of consisting of formed by which are meromorphic starlike of order in . In particular, we denote by , when . Also, a function of the form (1.9) is said to be meromorphic close-to-convex in if there is a in such that We denote the set of functions close-to-convex in .
In a recent paper, Gao and Zhou  introduced an interesting subclass of the analytic function class and the univalent function , which contains the functions satisfying the following inequality: for some . Here, is the class of starlike functions of order in . After that, many classes related to investigated and studied by some authors. Especially, Wang et al. [11, 12], Kowalczyk and Leś-Bomba , Xu et al. , and Seker  introduced the generalized class of the class . And they gave some properties of analytic functions in each classes.
In this paper, we define a class of meromorphic functions of Janowski’s type, related to the meromorphically close-to-convex functions as follows.
Definition 1.1. Let and be given by (1.9) and . Then , if there exists such that for ,
Theorem A. If , then .
Proof. At first, we know that for , since . And let . Then So we have Hence .
Therefore, we know that is a subclass of , by Theorem A.
We will obtain some coefficient bounds and some argument and convolution properties belonging to this class.
2. On the Coefficient Estimates of Functions in
Now we give the coefficient estimates of functions in . We need the following Lemma to estimate of coefficient of functions in .
Lemma 2.1. Let and be given by (1.11), then one has
Proof. According to the Theorem A, we have , and if let we know that , so is an odd meromorphic starlike function. If we let it’s well-known [16, Volume 2, page 232] that Substituting the series expressions of , in (2.2) and comparing the coefficients of two side of this equation, using (2.3) we can get an equality: and the proof of Lemma 2.1 is completed.
Theorem 2.2. Let given by (1.9) and . If then .
Proof. Let the functions and be given by (1.9) and (1.11), respectively. Furthermore, we let . Then where is given by (2.5) and . Now we obtain Thus, for , we have from (2.6) Thus we have which is equivalent to which implies that .
Proof. Let . Then we have where is an analytic function in , for and . Then, Thus, putting we obtain With comparing the coefficients, we can write Then, squaring the modulus of the both sides of the above equality and integrating along and using the fact the , we obtain Letting on the both sides, we obtain which implies the inequality (2.12).
Lemma 2.4 (see [2, Ruscheweyh and Sheil-Small]). Let and be convex in and suppose . Then .
Theorem 2.5. If , then there exists such that for all and with and ,
Proof. By definition, for , there exist and such that , Put , then And (2.22) implies that Since , hence For and such that , , the function is convex in . Applying Lemma 2.4 with this , we have Given any function analytic in with , we have so (2.27) reduces to Exponentiating both sides of (2.29) leads to (2.20).
3. Argument and Convolution Properties of Functions in
In this section, we will solve some problems related to argument and convolution properties of functions in the class . To solve these problems, we need the following Lemmas.
Lemma 3.2 (see [1, Ruscheweyh]). Let , and . Then for any analytic function , where denote the closed convex hull of .
Theorem 3.3. Let . Then
Proof. Let . Then there exists a function satisfying where And is contained in the disk from which it follows that Hence we can obtain the inequality: Hence it suffices to find the upper bounds of . Since , we have for and . If we define , then and we can apply Lemma 3.1 to obtain From the relation between and we obtain which implies Hence for , and so the result is proved.
Theorem 3.4. Let satisfying the condition and with . Then .
Proof. Let such that and define the functions and by Thus for any fixed , From the condition (3.13), we can easily obtain that and this inequality implies that . Put . Since is an odd meromorphic starlike function, so is. Define a function by . Since is an odd meromorphic starlike function, and . Hence Since and , Lemma 3.2 yields where denote the closed convex hull of , and because and is convex in , it follows that in . Thus and the proof of Theorem 3.4 is completed.
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