Abstract
For , , we consider the of normalized analytic convex functions defined in the open unit disc . In this paper, we investigate the class , that is, , with is Koebe type, that is, . The subordination result for the aforementioned class will be given. Further, by making use of Jack's Lemma as well as several differential and other inequalities, the authors derived sufficient conditions for starlikeness of the class of -fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in the earlier works are also indicated.
1. Introduction
Let denote the class of normalized analytic functions of the form which are analytic in the open unit disk . Also, as usual, let be the familiar classes of starlike functions in and convex functions in , respectively.
If the functions and are analytic in , then we say that the function is subordinate to , or is superordinate to (written as ) if there exist a function analytic , such that and , and with in . If is univalent in , then is equivalent to and .
Next, we let the , that is, The class was first introduced by Mocanu [1], which was then known as the class of convex (or -starlike) functions. Later, Miller et al. [2] studied this class and showed that is a subclass of for any real number and also that is a subclass of for . We note that and . Note also that Mocanu introduced with . But Sakaguchi and Fukui [3] later showed that this condition was not needed.
Motivated essentially by the aforementioned earlier works, we aim here at deriving sufficient conditions for starlikeness of -fold symmetric function of the Koebe type, defined by which obviously corresponds to the familiar Koebe function when and .
Definition 1.1. A function given by (1.1) is said to be in the class , for , , if the following conditions are satisfied:
In this paper, we consider the class of functions .
In addition, in this paper, authors investigate the subordination of the class denoted by .
We have the following inclusion relationships:(i)(ii), which has studied by [4].
The work of Siregar et al. [5] and Bansal and Raina [6] have also motivated us to come to these problems. Look also at [7, 8] for different studies.
The following result (popularly known as Jack's Lemma) will also be required in the derivation of our result (Theorem 4.1 below).
2. Preliminaries
Lemma 2.1 (see [9]). Let be univalent in and let the function and be analytic in a domain containing , with when . Set and suppose that (i) is univalent and starlike in ;(ii). If is analytic in with , , and then and is the best dominant.
Lemma 2.2 (see [10]). Let the (nonconstant) function be analytic in such that . If attains its maximum value on circle at a point , we have where is a real number.
3. The Subordination Result
Theorem 3.1. Let satisfy . Also, let the function be univalent in , with and , for and , such that If where then and is the best dominant of (3.2).
Proof. We first choose
then and are analytic inside the domain , which contains , , and when .
Now, if we define the functions and by
then it follows from (3.1) that is starlike in and
We also note that the function is analytic in , with . Since , therefore, , and hence, the hypothesis of Lemma 2.1 are satisfied.
Applying Lemma 2.1, we find that
which implies that
and is the best dominant of (3.2).
4. The Properties of the Class
We begin by proving a stronger result than what we indicated in the preceding section.
Theorem 4.1. Let the -fold symmetric function , defined by (1.4), be analytic in U, with
If satisfies the inequality:
, then is starlike in for
If satisfies the inequality (4.2) with , that is, if
then is starlike in for
Proof. Let , and satisfy the hypothesis of Theorem 4.1. We put
where is analytic in , with
such that, we can write
which, in turn, implies that
Now, we claim that . If there exists a in such that , then (by Jack’s Lemma) Lemma 2.2, we have
where is a real number.
By setting , thus, we find that
since .
If we let
then
Thus, we have
which is a contradiction to the hypotheses of (4.2).
Therefore, for all in . Hence is starlike in , then by proving the assertion (i) of Theorem 4.1, this completes the proof of our theorem.
Next, we arrive to the following remark which was given by Fukui et al. [11], and so we omit the detail here.
Remark 4.2. Let the -fold symmetric function , defined by (1.4), be analytic in , with If satisfies the inequality (4.2) with , that is, if then is starlike in for .
The following remark was obtained by Kamali and Srivastava [12].
Remark 4.3. Let the -fold symmetric function , defined by (1.4), be analytic in , with If satisfies the inequality (4.2) with , that is, if then is starlike in for
5. Applications of Differential Inequalities
We apply the following result involving differential inequalities with a view to deriving several further sufficient conditions for starlikeness of the -fold symmetric function defined by (1.4).
Lemma 5.1 (Miller and Mocanu [13]). Let be a complex-valued function such that being (as usual) the complex plane, and let Suppose that the functions satisfies each of the following conditions. (i) is continuous in .(ii) and .(iii)for allsuch that Let be analytic (regular) in such that If then
Let us now consider the following implication.
Theorem 5.2. Let -fold symmetric function , defined by (1.4) and analytic in with , satisfy the following inequality: then
Proof. If we put then (5.8) is equivalent to By setting and and letting for and , we have the following. (i)is continuous in.(ii)and since Thus, the conditions (i) and (ii) of Lemma 5.1 are satisfied. Moreover, for and , we obtain which, upon putting , yields where
Remark 5.3. If, for some choices of the parameters , , , and , we find that then we can conclude from (5.16) and Lemma 5.1 that the corresponding implication (5.8) holds true.
First of all, for the choice: and , we have the following.
Theorem 5.4. If -fold symmetric function , defined by (1.4) and analytic in with satisfies the following inequality: then for any real and .
Proof. For , , we find from (5.17) that which implies Theorem 5.4 in view of the remark.
Remark 5.5. For , we will obtain the results by Kamali and Srivastava [12].
Acknowledgment
The work presented here was supported by UKM-ST-06-FRGS0244-2010.