Certain Properties of Some Families of Generalized Starlike Functions with respect to q-Calculus

and Applied Analysis 3 0 1 2 3 4 5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5


Introduction and Preliminaries
The quantum calculus, so called -calculus and ℎ-calculus, is the usual calculus without using the notion of limits.The letter ℎ 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][4][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][7][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 Uc ¸ar [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][15][16][17][18][19][20][21]).
Let D  = { ∈ C : || < } be the open disk radius  centered at origin and the open unit disk is then defined by D ≡ D 1 .We denote A by the class of functions  in the form which is analytic in D and satisfying the usual normalization condition (0) =   (0) − 1 = 0. We denote by S the subclass of A consisting of functions, which are univalent on D. A function  ∈ A is said to be starlike of order  (0 ≤  < 1) We denote this class by S * ().In particular, we set S * (0) ≡ S * for a class of starlike functions on D.
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  ⊂ C 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 C 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  → 1 − and  ∈ N, 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.
To put it in words, we call S * ,1 () the class of -starlike functions of order  type 1.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 { : Re  > } 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.
To put it in words, we call S * ,2 () the class of -starlike functions of order  type 2.
In addition, we now introduce new type of -starlike functions.
To put it in words, we call S * ,3 () the class of -starlike functions of order  type 3.
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.
The next result is directly obtained by using Theorem 4 and the result in [14].

Functions with Negative Coefficients
Now, we introduce new subclasses of -starlike functions with negative coefficients.Let T be a subset of A containing negative coefficient functions; that is, Next, we let Theorem 8.For 0 <  < 1, then Proof.By using Theorem 4, it is sufficient to show that TS * ,1 () ⊂ TS * ,3 ().Assuming that  ∈ TS * ,1 (), we have Take  on the real axis so that the value of   ()/() is real.Letting  approach 1 − 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 TS * , () ≡ TS *  (), for  = 1, 2, and 3.By using Theorem 6, it is easy to see that function where That is,  0 () ∉ S and also  0 () ∉ S * ().So, it is interesting to study the radius of univalency and starlikeness of class TS *  ().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. and Proof.To prove this, we need to find 0 <  0 ≤ 1 such that Re{  ()} > 0 on D  0 , where D  0 = { ∈ C : || <  0 } due to the following formula: which implies the univalency.Consider for all || <  0 .By the application of Theorem 6 and ( 25), the inequality Re{  ()} > 0 holds on D  0 , where .
Differentiating on both sides of (27) logarithmically, we have where then the third and the last term in (28) can be dominated by ln  when  is sufficiently large.That implies that  is an increasing function on [ 0 , ∞], where  0 >  1+| ln(/(+))| .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 . and Proof.We have to show that |  ()/ − 1| < 1 − .By an application of Theorem 6, the above inequality holds on D  1 , where .

Examples and Applications
In this section, we give some examples to verify the radius of univalency and starlikeness obtained by Theorems 10 and 11.
The next example is the class of -starlike functions of order .
Next, we give a sufficient condition of S * ,3 via coefficient inequality which guarantees a sufficient condition for S * and only if  is starlike on D  .