Sufficiency Criteria for q-Starlike Functions Associated with Cardioid

School of Electrical Engineering and Computer Science (SEECS), National University of Sciences & Technology (NUST), Sector H-12, Islamabad 44000, Pakistan Department of Mathematics, University of Wah, Quaid Avenue, Wah Cantt, Pakistan NUST Institute of Civil Engineering (NICE), School of Civil and Environmental Engineering (SCEE), National University of Sciences & Technology (NUST), Sector H-12, Islamabad 44000, Pakistan Department of IT and Computer Science, Pak-Austria Fachhochschule Institute of Applied Sciences and Technology, Haripur, Pakistan Department of Mathematics, COMSATS University Islamabad, Wah Campus, Wah Cantt, Pakistan Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan


Introduction
Consider the class A of analytic functions defined in open unit disk F with normalization condition f ð0Þ = 0 and f ′ ð0Þ = 1 which provides the Taylor series expansion of the form f z ð Þ = z + 〠 ∞ n=2 a n z n , z ∈ F: The class S consists of functions from A which are univalent functions in F, and the class P contains the analytic functions whose codomains are bounded by the open right half plane. For more details, see [1,2]. The concept of differential subordination plays a vital role in the study of geometric properties of analytic func-tions. It was first introduced by Lindelof, but Littlewood [3] did the remarkable work in this field. Many researchers contributed in the study of differential subordinations. History and the development of works in the field related to differential subordination are briefly described and included in the book by Miller and Mocanu [4]. The major development in the field of differential subordination started in 1974 by Miller et al. [5].
An analytic function f is considered to be subordinated by analytic function g, denoted as f ≺ g, if there exists another analytic function w with the property that wð0Þ = 0 and jwðzÞ j < jzj such that f ðzÞ = gðwðzÞÞ: Moreover, in case of univalent functions in F, we can have Recently, many mathematicians have used this concept of differential subordinations to prove many helpful results. Familiar Jack's lemma [6] has produced several advancements for the generalization of differential subordinations and found many applications in this field. The work of Ma and Minda [7] in this field is not negligible as they studied the function Φ which is analytic, and condition of normalization given for prescribed function is defined as Φð0Þ = 1 and Φ′ð0Þ > 0 with a positive real part. With the help of the function Φ, they introduced the following subclasses for starlike and convex functions.
These subclasses helped many researchers for further studies in the field of differential subordination. Ali et al. [8] used the concept of differential subordination to prove analytic functions to be Janowski starlike. Ali et al. [9] also evaluated several differential subordinations: 1 + γzðp ′ ðzÞ/ p n ðzÞÞ and found the γ for pðzÞ ≺ ffiffiffiffiffiffiffiffiffi ffi 1 + z p . Raina and Sokol [10] used subordinations for coefficient estimation of starlike functions. Similar kinds of works have also been done by Sharma et al. [11] by using starlikeness for cardioid function, and Yunus et al. [12] studied for limacon.
Quantum calculus is the new branch of mathematics and is equally important for its applications both in physics and in mathematics as well. Jackson [13,14] presented the functions of q-derivatives and q-integrals and highlighted their definitions for the first time. He also holds the credit for the systematic initiation of q-calculus. Ismail et al. [15] were the pioneers to contribute in the application of q-calculus in geometric function theory. The new form of the subclass of starlike functions S * ðΦÞ with the involvement of q-derivative was introduced by Seoudy and Aouf [16]. By choosing different image domains instead of ΦðzÞ, so many attractive subclasses of starlike functions are obtained. Mahmood et al. [17] have dealt with the class of q-starlike functions by relating them with conic domains. The most recent work related to q -starlikeness of functions is done by Srivastava et al. [18]. The contributions of Haq et al. [19] are remarkable. They proved differential subordinations with q-analogue for cardioid and limacon domain with the involvement of Janowski function and found the sufficient conditions for q-starlike functions. The q version of Jack's lemma which is the soul of our work was given by Çetinkaya and Polatoglu [20]. These recent efforts of mathematicians discussed above motivated us and provide strength to contribute in the field of differential subordinations with the involvement of its q-analogue, which is the main idea of this article. The foundation of all this work in q-analogue is the q-derivative which is defined below.
The q-derivative of a complex-valued function f , defined in the domain F, is given as follows: where 0 < q < 1: This implies the following: provided the function f is differentiable in domain F: The function D q f has Maclaurin's series representation where For more details about q-derivatives and recent work on it, we refer the reader to [21][22][23][24][25].
Lemma 2 (q-Jack's lemma, [20]). Consider an analytic function w in F with wð0Þ = 0. For a maximum value of w on the circle jzj = 1 at z 0 = ae iθ , where θ ∈ ½−π, π, and 0 < q < 1, then, we have Here, m is real and m ≥ 1: By using the above lemma, we have proved our main results.

Main Results
and we define an analytic function h on F with hð0Þ = 1 which satisfies 2 Journal of Function Spaces In addition, we suppose that where w is analytic in F with wð0Þ = 0: Then, Proof. Consider the function which is analytic in F with the condition pð0Þ = 1 and the function where w is an analytic function in F with wð0Þ = 0: To prove the result, it would be sufficient to show that jwðzÞj ≤ 1 for From (14) and (15), we deduce the following: and with this, one can have This implies that Now, considering the existence of a point z 0 ∈ F such that Now, we use the q-Jack's lemma which implies that there exist a number m ≥ 1 such that z 0 ∂qwðz 0 Þ = mwðz 0 Þ: This, with the consideration that wðz 0 Þ = e iθ , θ ∈ ½−π, π for z 0 ∈ F, we have The function 3 Journal of Function Spaces is clearly an even function. So, in order to find the maximum value of G, we will consider the interval ½0, π: Thus, gives G ′ ðθÞ = 0 for θ = 0 and π. Also, we can see that G ′ ′ðπÞ > 0 for 1 < m < 2:5, which results that GðθÞ ≥ GðπÞ: Now, consider the function Thus, ΘðmÞ is an increasing function which gives a minimum value for m = 1: Then, we have From (10), we conclude that but this result contradicts (11). Hence, jwðzÞj < 1 and this leads us to the desired result. By taking hðzÞ = z∂ q f ðzÞ/f ðzÞ, the above result reduces to the following.

Corollary 4. Let γ ≥ 3ð
ffiffi ffi 2 p + 1Þ/2ð1 − qÞ and f ∈ A satisfy the subordination Then, f ðzÞ ∈ S * q,c : and we define an analytic function h on F with hð0Þ = 1 which satisfies In addition, we suppose that where w is analytic in F with wð0Þ = 0: Then, Proof. Consider the function which is analytic in F with the condition pð0Þ = 1 and the function Journal of Function Spaces where w is an analytic function in F with wð0Þ = 0: Using (33) and (34), we obtain Proving the fact that jwðzÞj ≤ 1 will be sufficient to prove our assertion. For this, consider Considering the existence of a point z 0 ∈ F such that we can make use of q-Jack's lemma which implies that there exists a number m ≥ 1 such that z 0 ∂qwðz 0 Þ = mwðz 0 Þ: Now, consider that wðz 0 Þ=e iθ , θ ∈ ½−π, π, then for z 0 ∈ F, we have is clearly an even function. So, in order to find the maximum value of G, we will consider the interval ½0, π: Now, we have for θ = 0 and π: Also, we can see that G ′ ′ðπÞ > 0 for m≥1, thus we conclude that GðθÞ ≥ GðπÞ: So we have the function This gives Thus, ΘðmÞ is an increasing function which gives a minimum value for m = 1. Then, we have From (29), we conclude that but this result contradicts (30). Hence, jwðzÞj < 1 which provides the required result. By taking hðzÞ = z∂ q f ðzÞ/f ðzÞ, the above result reduces to the following. Corollary 6. Let γ ≥ ffiffi ffi 2 p + 1/2ð1 − qÞ and f ∈ A satisfy the subordination Then, f ðzÞ ∈ S * q,c : and we define an analytic function h on F with hð0Þ = 1 which satisfies In addition, we suppose that Proof. Let us define the function which is analytic in F with the condition pð0Þ = 1 and the function where w is an analytic function in F with wð0Þ = 0: Using (51) and (52) To prove the assertion, it would be enough to show that jwðzÞj ≤ 1: Therefore,