Abstract
By making use of the concept of -calculus, various types of generalized starlike functions of order were introduced and studied from different viewpoints. In this paper, we investigate the relation between various former types of -starlike functions of order . We also introduce and study a new subclass of -starlike functions of order . Moreover, we give some properties of those -starlike functions with negative coefficient including the radius of univalency and starlikeness. Some illustrative examples are provided to verify the theoretical results in case of negative coefficient functions class.
1. Introduction and Preliminaries
The quantum calculus, so called -calculus and -calculus, is the usual calculus without using the notion of limits. The letter apparently stands for Planck’s constant and the letter obviously stands for quantum. Here, quantum calculus is not the same as quantum physics. Due to the applications in various fields of mathematics and physics, the study of -calculus has been very attractive for many researchers. Jackson [1, 2] was the first person in developing a -derivative, also a -integral, in a systematic mean. Afterward on quantum groups, the geometrical interpretation of -analysis has been studied. The relation between -analysis and integrable systems has been recognized. Based on -analogue of beta function, Aral and Gupta [3–5] defined and studied the -analogue of Baskakov Durrmeyer operator. Also, there are some discussions on -Picard and -Gauss-Weierstrass singular integral operators which are the other important -generalization of complex operators (see [6–8]).
In geometric function theory, there are many applications of -calculus on subclasses of analytic functions, especially subclasses of univalent functions. In [9], Ismail et al. first introduced the class of generalized functions via -calculus. In [10], Raghavendar and Swaminathan have studied some basic properties of -close-to-convex functions. In [11], Mohammed and Darus studied geometric properties and approximations of these -operators in some subclasses of analytic functions in the disk. By using the convolution of normalized analytic functions and -hypergeometric functions, these -operators have been defined. The inclusive study on applications of -calculus in operator theory could be seen in [12]. Recently, Esra Özkan Uçar [13] studied the coefficient inequality for -closed-to-convex functions with respect to Janowski starlike functions. Here, many newsworthy results related to -calculus and subclasses of analytic functions theory are studied by various authors (see [14–21]).
Let be the open disk radius centered at origin and the open unit disk is then defined by . We denote by the class of functions in the formwhich is analytic in and satisfying the usual normalization condition . We denote by the subclass of consisting of functions, which are univalent on . A function is said to be starlike of order in if satisfiesWe denote this class by . In particular, we set for a class of starlike functions on . Class is closely related to class ). A function is said to belong to class if satisfies
For the convenience, we provide some basic definitions and concept details of -calculus which are used in this paper. For any fixed complex number , a set is called a -geometric set if for , . Let be a function defined on a -geometric set. Jackson’s -derivative and -integral of a function on a subset of are, respectively, given by (see Gasper and Rahman [22], pp. 19–22)In case , the -derivative and -integral of , where is a positive integer, are given by As and , we have .
To generalize the class of starlike functions, it seems that replacing the derivative function , which appears in (2), by the -difference operator is an easily way to generalize the class of starlike functions. The definition turned out to be the following.
Definition 1. A function is said to belong to class , , ifTo put it in words, we call the class of -starlike functions of order type .
Now we recall another way to generalize the class of starlike functions proposed by Ismail et al. [9]. In their works, the usual derivative was replaced by the -difference operator . Moreover, the right-half plane was substituted by an appropriate domain. Later, Agrawal and Sahoo in [14] extended the ideas in [9] to -starlike function of order . Then the definition turned out to be the following.
Definition 2. A function is said to belong to class , , ifTo put it in words, we call the class of -starlike functions of order type .
In addition, we now introduce new type of -starlike functions.
Definition 3. A function is said to belong to class , , ifTo put it in words, we call the class of -starlike functions of order type .
The main objective of this paper is to characterize in 4 sections. In Section 2, we give some relations between such classes and a sufficient condition via coefficient inequality. In Section 3, we study some properties of those -starlike functions of order with negative coefficient. Here, some results on the radius of univalent and starlikeness order on the class of -starlike functions with negative coefficient are obtained. Some illustrative examples of radius of univalent and starlikeness on some functions with negative coefficient are demonstrated in Section 4.
2. Main Results
We first show the inclusion theorem via geometric properties of each type of -starlike functions.
Theorem 4. For , then
Proof. Assuming that , by using triangle inequality and (8), we haveThen ; that is, . Next, we let . Since that is, lies in the circle of radius with a center at , and we observe that which means that , then ; that is, . This completes the proof.
Geometrically, for , , lied in the difference domains:respectively; see Figure 1.
The next result is directly obtained by using Theorem 4 and the result in [14].
Corollary 5. Classes , , and satisfy the following properties:
Next, we give a sufficient condition of via coefficient inequality which guarantees a sufficient condition for and .
Theorem 6. If satisfies the inequalitythen is a -starlike function of order type ; that is, .
Proof. Suppose that inequality (15) holds. We obtainThen as desired.
Remark 7. In Theorem 6, if , we obtain Theorem 1 in [23].
3. Functions with Negative Coefficients
Now, we introduce new subclasses of -starlike functions with negative coefficients. Let be a subset of containing negative coefficient functions; that is, Next, we let
Theorem 8. For , then
Proof. By using Theorem 4, it is sufficient to show that . Assuming that , we haveTake on the real axis so that the value of is real. Letting approach on the real line, we have which satisfies (15). Theorem 6 implies the proof of this theorem.
By using the result of Theorem 8, all types of -starlike functions are exactly the same. For convenience, we introduce a new notation for each class of -starlike functions , for , and
By using Theorem 6, it is easy to see that function where and , but at . That is, and also . So, it is interesting to study the radius of univalency and starlikeness of class .
Lemma 9 is required to prove the radius of univalency and starlikeness. By using the same techniques of Theorem 1 in [24] and Theorem 1 in [25], we can easily prove Lemma 9. So, the proof is omitted.
Lemma 9. If , then is univalent on if and only if is starlike on .
Theorem 10. If then is univalent and starlike in , whereand satisfies .
Proof. To prove this, we need to find such that on , where due to the following formula:which implies the univalency. Considerfor all . By the application of Theorem 6 and (25), the inequality holds on , where Next, we need to find satisfying (23). Let be the function defined byDifferentiating on both sides of (27) logarithmically, we havewhere . It is easy to see that the second term of (28) is positive. Since then the third and the last term in (28) can be dominated by when is sufficiently large. That implies that is an increasing function on , where . Therefore, the radius of univalency can be defined by Finally, we complete the proof of this theorem by applying Lemma 9 to obtain the radius of starlikeness.
Theorem 11 guarantees the radius of starlike function of order .
Theorem 11. If then is starlike order in , whereand satisfies .
Proof. We have to show that . That is,Hence, (32) is true if By an application of Theorem 6, the above inequality holds on , where Finally, by using the same technique of Theorem 10, we obtain that function is an increasing function on , where satisfies . This completes the proof.
4. Examples and Applications
In this section, we give some examples to verify the radius of univalency and starlikeness obtained by Theorems 10 and 11.
Example 1. Consider class with .
By Theorem 10, we obtain the radius of univalency of class given by Now, we consider the sharpness example function defined in (22) with and ; that is,Obviously, is locally univalent on because at outside the open disk . By applying Theorem 10, function is univalent on . Moreover, Figure 2 shows the image of with maximum circumferences and . Figure 2(a) demonstrates that function is a univalent and starlike function on . On the other hand, is not a univalent on (see Figure 2(b)).
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(b)
Another example is in case with ; that is,We see that is not locally univalent at , for with . Figure 3 shows that function defined in (37) is univalent and starlike on which contains the open disk from Theorem 10. That is, the example shows that radius in Theorem 10 is only the sufficient condition for univalency and starlikeness but it is not necessary condition due to function defined in (37).
The next example is the class of -starlike functions of order .
Example 2. Consider class with .
In this example, we also set . For , by Theorem 10, we obtain the radius of univalency of class given byHowever, function defined in (22) with and , that is,is locally univalent on which contains the open disk . Then it seems that function is univalent and starlike on as demonstrated by Figure 4(a). Also function defined in (22) with and , that is,is locally univalent on and it seems that function is univalent and starlike on as demonstrated by Figure 4(b).
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(b)
Competing Interests
The authors declare that they have no conflict of interests.
Acknowledgments
This research was supported by Department of Mathematics, Faculty of Science, Chiang Mai University.