Q -Extension of Starlike Functions Subordinated with a Trigonometric Sine Function

: The main purpose of this article is to examine the q -analog of starlike functions connected with a trigonometric sine function. Further, we discuss some interesting geometric properties, such as the well-known problems of Fekete-Szegö, the necessary and sufﬁcient condition, the growth and distortion bound, closure theorem, convolution results, radii of starlikeness, extreme point theorem and the problem with partial sums for this class.


Introduction and Definitions
To understand all the concepts used in this article clearly we need to include and explain all the terms mentioned here. First, let A be the collection of functions which are holomorphic (or analytic) in D := {z ∈ C : |z| < 1} and fulfill the subsequent Taylor series expansion: In [1,2], Miller and Mocanu generalized the ideas that consist of differential inequalities for real to complex valued functions that laid the foundations for a new theory, known as "the method of differential subordination or admissible functions method". This technique is used in geometric function theory, as a tool that provides not only new results, but also solves complicated problems in a simple way. In complex valued function the characterization of a function can be obtained from a differential condition, for example, the Noshiro-Warschawski theorem [3]. Said theory is applicable in various fields, including ordinary differential equations, partial differential equations, harmonic functions, integral operators, Banach spaces and functions of several variables.
If f 1 and f 2 is in A, then f 1 is subordinated by f 2 if a holomorphic function w can be find with the properties w (0) = 0 and |w (z)| < |z| so that f 1 (z) = f 2 (w(z)) (z ∈ D). In addition, if f 1 , f 2 ∈ D are univalent, then: Additionally, the Hadamard product (or convolution) between the functions f 1 , f 2 ∈ A is described by In 1994, Ma and Minda [4] introduced the following subset of holomorphic functions: with the restriction that the image domain of h (h is a convex function with Reh > 0 in D) is symmetric along the real axis and starlike about h(0) = 1 with h (0) > 0. They investigated certain useful problems, including distortion, growth and covering theorems. Now taking some particular functions instead of h in S * (h), we achieve many sub-families of the collection A which have different geometric interpretations as for example: is the set of Janowski starlike functions; see [5]. Some interesting problems such as convolution properties, coefficient inequalities, sufficient conditions, subordinates results and integral preserving were discussed recently in [6][7][8][9][10] for some of the generalized families associated with circular domains.

(ii)
The class S * L := S * ( √ 1 + z) was introduced by Sokól and Stankiewicz [11], consisting of functions f ∈ A such that z f (z)/ f (z) lies in the region bounded by the right-half of the lemniscate of Bernoulli given by |w 2 − 1| < 1.
Recently in 2019, Cho and his coauthors [20] established the following class S * sin by selecting the function 1 + sin z instead of the function h as: Geometrically, the ratio f (z) lies in an eight-shaped region in the right half plane. They investigated the inverse inclusion relations of this family with the already known subfamilies of analytic functions. Later on for this family, the third Hankel determinants were studied by the authors in [21].
The classical calculus with no limit is known as quantum calculus or just q-calculus. This exceptional theory emerged via Jackson [22,23]. The readers were influenced by the q-calculus learning owing to its contemporary usage of numerous arguments as for example; in quantum theory, special functions theory, differential equations, number theory, operator theory, combinatorics, numerical analysis and certain other similar theories; see [24,25]. The early work of q-calculus in the field of geometric function theory (GFT) was done by Ismail et al. (see [26]) by generalizing the set of starlike functions into a q-analogue, known as the set of q-starlike functions. Another important development in this direction was the work of Anastassiu and Gal [27,28], who gave the q-generalizations of certain complex operators (particularly Picard and Gauss-Weierstrass singular integral operators). Following the same idea, Srivastava [29] presented some strong footing by giving some applications of q-calculus in this field by using q-analogues of hypergeometric functions. In this direction, some good valuable contributions were made by researchers, including Srivastava [30], Agrawal [31], Seoudy and Aouf [32], Agrawal and Sahoo [33], Arif and Ahmad [34], Kanas and Rȃducanu [35], Arif, Srivastava and Umar [36] and Haq et al. [37]. See also the articles [38][39][40][41][42][43].
For q ∈ ]0, 1[ and z ∈ D, the q-analog derivative of f is defined by If we take f (z) = ∑ ∞ n=1 a n z n , then for n ∈ N (natural number set) and z ∈ D [n] q a n z n−1 , Using the above mentioned concepts, we now define the following family S * sin (q) of starlike functions by: We note that lim q→1 − S * sin (q) ≡ S * sin , the class given by Equation (3). In this paper, we study some essential properties, such as the inequality of Fekete-Szegö, convolution problems, necessary and sufficient conditions, coefficient inequality, growth and distortion bounds, closure theorem, extreme point theorem and the partial sums problem.
The following two lemmas are used in the paper. However, before the statements of lemmas we define the class P of functions with a positive real part.
Let P denote the family of all functions p 1 that are analytic in D with positive real parts and have the following series representation: Lemma 1. [4] If f ∈ P has the expansion form given in Equation (7), then for ϑ ∈ C,

Major Contributions
Theorem 1. Let f ∈ S * sin (q) have the representation given in Equation (1). Then for ϑ ∈ C Proof. Let f ∈ S * sin (q). Then one can conveniently write Equation (6) in terms of the Schwarz function w as Additionally, if p 1 ∈ P, then Alternatively From Equations (1) and (5), we easily have and on the other hand From the last two equations, we get Now using Equations (13) and (14), we obtain By applying Lemma 1 to Equation (15) we get hence, proof is complete.
If we put λ = 1, in Theorem 1, we deduce the result below.
By making q → 1 − in Theorem 1, we achieve: [21] Let f ∈ S * sin . Then for ϑ ∈ C Theorem 2. Let f ∈ S * sin (q) and is of the form given by Equation (1). Then Proof. Using Equations (13) and (14), we have Using Lemma 2 to Equation (16), we obtain the required result.

Theorem 4. A necessary and sufficient criteria for a holomorphic function f ∈ S *
[n] q − 1 + sin e iθ sin e iθ a n z n−1 = 1.
Proof. In the light of above Theorem 3, we have f ∈ S * sin (q) if and only if Using series form of f and z∂ q f , we have [n] q a n z n − H ∞ ∑ n=2 [n] q − 1 a n z n sin e iθ a n z n−1 ; hence the relation (20) is proved.
Proof. In order to establish this theorem, we use relation (20). We have [n] q − 1 + sin e iθ sin e iθ a n z n−1 > 1 − ∞ ∑ n=2 [n] q − 1 + sin e iθ sin e iθ a n z n−1 |a n | z n−1 [n] q − 1 + sin e iθ sin e iθ |a n | > 0, and hence by virtue of Theorem 4, the proof is completed.
Theorem 6. Let f ∈ S * sin (q), and |z| = r. Then Proof. Consider On other hand, Theorem 7. Let f ∈ S * sin (q), and |z| = r. Then Proof. The proof is similar to that of Theorem 6 and it is omitted.

Proof. We have
δ k a n,k z n .
Theorem 9. The class S * sin (q) is a convex set.
The relation (23) is bounded by 1 if and hence the proof is completed.

Partial Sum Problems
In this section, we examine the partial sum problems of certain analytic functions contained in the family S * sin (q). We produce some new findings that have a connection between the analytical functions and their partial sum sequences. If a function f ∈ A has the series form given in Equation (1), then the partial sum f m of f is described by f m (z) = z + m ∑ n=2 a n z n with f 1 (z) = z.
In 1928, Szegö [44] proved an interesting result which states that if f ∈ S * , then This result motivated researchers to study the problem of partial sums for sub-families of analytic, univalent and multivalent functions. In [45], Silverman determined sharp lower bounds on the real parts of the quotients between the normalized convex or star-like functions and their consequences of partial sums. Additionally, Singh [46], Shiel-Small [47], Robertson [48], Ruscheweyh [49], Ponnusamy et al. [50], Srivastava et al. [25] and Owa et al. [24], have derived some beautiful results involving the partial sums.
The above given results are the best ones.
Proof. To prove relation (26), let us write a n z n−1 1 + ∑ m n=2 a n z n−1 where w (z) = d m+1 ∑ ∞ n=m+1 a n z n−1 2 + 2 ∑ m n=2 a n z n−1 + d m+1 ∑ ∞ n=m+1 a n z n−1 . Now Finally, to show relation (30), it is sufficient to establish that the left-side of relation (30) is bounded above by ∑ ∞ n=2 d n |a n | and it is equal to The last inequality is true because of relation (28). To show that the inequality (26) is sharp, let us consider the function Then for z = re i π m , we have To derive inequality (27), let us write a n z n−1 1 + ∑ m n=2 a n z n−1 where w (z) = − (1 + d m+1 ) ∑ ∞ n=m+1 a n z n−1 2 + 2 ∑ m n=2 a n z n−1 − (1 + d m+1 ) ∑ ∞ n=m+1 a n z n−1 . Now Finally, to obtain inequality (31), it is enough to show that the left side of inequality (31) is bounded by ∑ ∞ n=2 d n |a n | and is equivalent to which is true due to relation (28).

Conclusions
Utilizing the principle of subordinations, we have defined the family of q-starlike functions connected with a particular trigonometric function such as sine functions. The new class generalizes the class of starlike functions subordinated with sine function which was introduced by Cho et al. [20] in which the radii problems were investigated. For the newly defined class, we have first investigated the familiar Fekete-Szegö type problems. After that, we have proved some convolution results which were used in proving the necessary and sufficient condition for the defined class. The problem of partial sums has been established with the help of sufficiency criteria for this newly defined class. Some other problems, such as radii of starlikeness, closure theorem, growth and distortion bounds and extreme point theorem have also been studied here for this class. Moreover, the present idea can be extended to prove some other problems, such as the Hankel determinant, the sufficiency criterion and convolution conditions for this class. Furthermore, these results can also be obtained for starlike functions associated with cosine functions. This class was recently studied in [51].
Author Contributions: These authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.