Certain Class of Analytic Functions Connected with q-Analogue of the Bessel Function

Department of Mathematical Sciences, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia Department of Mathematics, GSS, GITAM University, Bengaluru Rural, Doddaballapur 562 163, Karnataka, India Research and Development Wing, Live4Research, Tiruppur 638 106, Tamilnadu, India Department of Electronics and Communication Engineering, Aditya College of Engineering and Technology, Surampalem 533 437, Andhra Pradesh, India Department of Mathematics, Kakatiya University, Warangal 506 009, Telangana, India


Introduction
Let A specify the category of analytic functions and η represent on the unit disc Δ � w: |w| < 1 { } with normalization η(0) � 0 and η ′ (0) � 1 such that a function has the extension of the Taylor series on the origin in the form Indicated by S, the subclass of A is composed of functions that are univalent in Δ.
en, a η(w) function of A is known as starlike and convex of order ϑ if it delights the pursing R wη ′ (w) η(w) > ϑ, w ∈ Δ, for specific ϑ(0 ≤ ϑ < 1), respectively, and we express by S * (ϑ) and K(ϑ) the subclass of A, which is expressed by the aforesaid functions, respectively. Also, indicated by T, the subclass of A is made up of functions of this form and let T * (ϑ) � T ∩ S * (ϑ), C(ϑ) � T ∩ K(ϑ). ere are interesting properties in the T * (ϑ) and C(ϑ) classes which were thoroughly studied by Silverman [1] and Alessa et al. [2]. e intense devotion of scientists has recently fascinated the study of the q-calculus. e great focus in many fields of mathematics and physics is due to its benefits. In the analysis of many subclasses of analytic functions, the importance of the q-derivative operator D q is very evident from its applications. e concept of q-star functions was originally proposed by Ismail et al. [3] in the year 1990. However, in the sense of Geometric Function eory, a firm basis of the use of the q-calculus was effectively developed. For example, in the rotational flow of Burge's fluid flowing through an unbounded round channel [3], it is used to derive velocity and stress.
After that, numerous mathematicians have carried out remarkable studies, which play a significant role in the development of geometric function theory. Furthermore, Srivastava [4] recently published a survey-cum-expository analysis article that could be useful for researchers and scholars working on these topics. e mathematical description and applications of the fractional q-calculus and fractional q-derivative operators in geometric function theory were systematically investigated in this surveycum-expository analysis article [4]. In particular, Srivastava et al. [5] also considered some function groups of conical region related q-star like functions. For other recent investigations involving the q-calculus, one may refer to [6][7][8][9][10][11][12][13].
One of the most significant special functions is the Bessel function. As a result, it is important for solving a wide range of problems in engineering, physics, and mathematics (see [14]). In recent years, several researchers have focused their efforts on forming different types of relationships. Many researchers have recently focused on determining the various conditions under which a Bessel function has geometric properties such as close-to-convexity, starlikeness, and convexity in the frame of a unit disc Δ. e first-order Bessel function ℘ is defined by the infinite series [15]: where Γ stands for a function of Gamma. Szasz and Kupan [16] and irupathi Reddy and Venkateswarlu [17] have recently explored the univalence of the first-kind normalization Bessel function k ℘ : Δ ⟶ C defined by For 0 < q < 1, El-Deeb and Bulboaca [18] defined the q-derivative k ℘ operator as follows: where Using (7), we are going to define two products in the text: (1) e q-shifted factorial is given for any nonnegative integer ]: (2) e q-generalized Pochhammer symbol for any positive number r is defined by For ℘ > 0, ℓ > − 1, and 0 < q < 1, El-Deeb and Bulboaca [18] defined J ℓ ℘,q : Δ ⟶ C by (see [19]) A simple computation shows that where the function M q,ℓ+1 (w) is supplied with the function El-Deeb and Bulboaca [18] introduced the linear operator using the definition of q-derivative along with the idea of convolutions N ℓ ℘,q : A ⟶ A defined by 2 Journal of Mathematics Remark 1 (see [18]). We can easily verify from the definition relation (13) that the next relation holds for all η ∈ A: Now, we propose a new subclass ϕ ℓ ℘,q (Z, ϑ) of A concerning qanalogue of the Bessel function as follows.

Coefficient Inequalities
is section gives us an adequate requirement for a function η given by (1) to be in ϕ ℓ ℘,q (Z, ϑ).

Remark 2. If a function η of form (3) belongs to the class
e equality holds for the functions

Distortion Theorem
In the section, the distortion limits of the functions are owned by the class Tϕ ℓ ℘,q (Z, ϑ).

Radii of Close-to-Convexity and Starlikeness
A close-to-convex and star-like radius of this class Tϕ ℓ ℘,q (Z, ϑ) is obtained in this section.
Proof. We have η ∈ T and η is order of starlike ℓ, and we have For the L.H.S of (39), we have We know that .

(43)
It yields starlikeness of the family.

Partial Sums
Silverman [20] examined partial sums η for the function η ∈ A given by (1) and established through In this paragraph, in the class ϕ ℓ ℘,q (Z, ϑ), partial function sums can be considered and sharp lower limits can be reached for the function. True component ratios are η to η m and η ′ to η m ′ . Theorem 7. Let η ∈ ϕ ℓ ℘,q (Z, ϑ) fulfil (16). en, where and it is good enough to show R(g(w)) > 0, w ∈ Δ. Applying (51), we think that which gives and hence proved.

Journal of Mathematics
where Proof. By setting the evidence is close to that of (51) and (52) theorems, so the specifics are omitted.

Convolution Properties
We will prove in this section that the Tϕ ℓ ℘,q (Z, ϑ) class is closed by convolution.
Employing the last inequality and the fact that we obtain and hence, in view of eorem 1, the result follows.

Conclusions
is research has introduced q-analogue of the Bessel function and studied some basic properties of geometric function theory. Accordingly, some results related to coefficient estimates, growth and distortion properties, convex linear combination, partial sums, radii of close-to-convexity 8 Journal of Mathematics and starlikeness, convolution, and neighborhood properties have also been considered, inviting future research for this field of study.

Data Availability
No data were used to support the findings of the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.