Certain Classes of Analytic Functions Bound with Kober Operators in q-Calculus

(rough applying the Kober fractional q-calculus apprehension, we preliminary implant and introduce new types of univalent analytical functions with a q-differintegral operator in the open disk U � ξ ∈ C:∣ξ|< 1 { }. (e coefficient inequality and distortion theorems are among the results examined with these forms of functions. Specific cases are responded and addressed immediately. (e findings include an expansion of the numerous established results in the q-theory of analytical functions.


Introduction and Preliminary
e q-analysis theory has been applied in recent times in several fields of science and engineering. e fractional q-calculus is indeed an analog of the conventional fractional calculus in q-theory. Very recently, Wang et al. [1] and Yan et al. [2] investigated the properties of subclasses of multivalent analytic or meromorphic functions expressed with q-difference operators. Furthermore, Srivastava [3] investigated the excellent work with q-calculus and fractional q-calculus operators, which is quite valuable for academics working on these issues. e applications of fractional q-calculus operators have been investigated by Purohit and Raina [4] to describe several new classes of analytic functions in open disk U � ξ ∈ C: |ξ| < 1 { }. Moreover, Murugusundaramoorthy et al. [5], Purohit [6], and Purohit and Raina [4,7] gave related work and added various classes of univalent and multivalently analytic functions in open unit disk U. Several others have also released new classes of analytical functions with the resources of q-calculus operators. For any more inquiries on the analytic functions classes, we refer to [1,2,[8][9][10][11][12][13] for functions described by applying q-calculus operators and subject related to this work.
In the current inquiry, we are planning to develop few additional families of analytic functions applying the Kober differential and integral operators in q-calculus. e results obtained must also provide the coefficient inequalities and distortion theorems for the subclasses established here below. First, we use the main notations and definitions in the q-calculus which are relevant to grasp the object of the study.
For each complex number P, the q-shifted factorials are delimited by and with regard to the basic analog of the gamma function, in which the q-gamma function is set by (see [14]) e recurrence relationship specified by Gaspar and Rahman [15] for the q-gamma function is If |q| < 1, then equation (1) shall continue to play a role m � ∞ as an infinite product of convergence: and we have e q-binomial expansion is now as follows: [15] accounts for Jackson's q-integral and q-derivative of a function f, which are described on a subset of C, as with (9)

The Fractional q-Calculus Operators
Purohit and Raina [4] described the fractional q-integral operator of function f(ξ) given by where P > 0 is the order of integral and f(ξ) is an analytic function in U, and (7) the (ξ − yq) P− 1 be expressed as where 1 ϕ 0 [P; − ; q, ξ] is special case of basic hypergeometric series 2 ϕ 1 [P; J; c; q, ξ] for c � J is single valued for |arg(ξ)| < π and |ξ| < 1 (see [15]). Purohit and Raina [4] defined the D P q,ξ f(ξ) fractional q-derivative operator of a function f(ξ) by where 0 ≤ P < 1 and f(ξ) is suitably constrained with e Kober fractional q-integral operator for a real valued function f(x) is determined by Garg and Chanchalani [16] as where c being real or complex and P is an absolute order of integration with R(P) > 0. For q ⟶ 1, operator (13) is reduced to Kober operator I c,P f(x) as defined in [17]. For c � 0, this operator is converted to Riemann-Liouville fractional q-integral operator with a power weight function . e Kober fractional q-derivative operator for a real valued function f(x) is detailed by Garg and Chanchalani [16] as where P is order of derivative with R(P) > 0 and m � [R(P)] + 1, m ∈ N. For q ⟶ 1, operator (14) is reduced to Kober operator D c,P f(x) as defined in [17].
We are now defining q-calculus operators with a view to applying these operators to the geometric function theory of complex analysis.

Definition 1. Kober fractional q-integral operator:
For the function f(ξ), the Kober fractional q-integral operator is demarcated by where c is the real or complex, P is an absolute order of integration with R(P) > 0, and the q-binomial (ξ − yq) P− 1 is expressed as in (11). For q ⟶ 1, operator (15) is reduced to Kober integral operator I c,P f(ξ) as defined in [17].

Definition 2. Kober fractional q-derivative operator:
e Kober fractional q-derivative operator for the function f(ξ) is demarcated by where P is the order of derivative with R(P) > 0 and m � [R(P)] + 1, m ∈ N. For q ⟶ 1, operator (16) is reduced to Kober derivative operator D c,P f(ξ) as defined in [17]. Under Kober q-integral and q-derivative operators fixed by (15) and (16), we offer the following image formulae for function ξ μ .

New Classes of Functions
Let A m represent the function class of the form which are analytic and univalent in open unit disk U. Above, let A − m highlight the subclass of A m imposing of analytical and univalent functions articulated in the form (20) For the dedication of this work, we describe a fractional q-differintegral operator Ω c,P q for a function f(ξ) of the form (20) by where and R(c + 2) > 0, 0 < q < 1, ξ ∈ U, m ∈ N, R(P) > 0, and D c,P q f(ξ) represent a fractional q-derivative of f(ξ) of order P. We announce here the alike classes of functions connecting operator (21): where e subsequent coefficient bounds for functions of the form (20) that belong to the classes S c,P q (T) and T c,P q (τ) are now obtained (interpreted above).

Theorem 1. A function f defined by (20) is connected to the class S c,P q (T) if and only if
where The result is sharp.
Proof. Let us consider that inequality (25) holds, and for |ξ| � 1, we have Journal of Mathematics and by our assumption, this indicates that f(ξ) ∈ S c,P q (T). For the proof of converse part, suppose that f(ξ) ∈ S c,P q (T), and then it follows that which implies that Since |R(ξ)| ≤ |ξ| for any ξ, therefore on choosing values of ξ on the real axis so that Ω c,P q f(ξ) is real and allowing ξ ⟶ 1 all through real values, we obtain from above inequality which is desired result. Here, we notice that assumption (25) of eorem 1 is sharp and the external function is assumed by where A(P, c, k, q) is defined in (26).

Theorem 2. A function f defined by (20) is connected to the class T c,P q (τ) if and only if
where The accomplishment is sharp.
Proof. To prove above theorem, we address the elementary assertion that Now, where 4 Journal of Mathematics (37) In (35), it then suffices to show that is accomplishes the proof of theorem. We accommodate that answer (33) is sharp. e external function is assumed by where A k,q (P, c, τ) is given by (34).

Distortion Theorems
Theorem 3. Suppose that the function f is defined by (20) in the class S c,P q (T), then where Furthermore,
Proof. Since f(ξ) ∈ S c,P q (T), then in interpretation of eorem 1, we first show that the function is an increasing function of k for c > − 2 and R(P) > 0. It follows that Taking k � m + 1, then e function ϕ(k) is an increasing function of k if (ϕ(m + 2)/ϕ(m + 1)) ≥ 1, and this gives which implies is inequality abides for R(P) > 0. us, ϕ(k), (k ≥ m + 1, m ∈ N) is an increasing function of k for R(c + 2) > 0, R(P) > 0 and 0 < q < 1. Now, (25) gives the alike inequality: where B(P, c, m, q) is defined in (42), and this last inequality is in the conjunction with the alike inequality (easily obtained from (20)): and using (50), we have which is result (41) of eorem 3. Now, on using (21), we observe that for functions of form (20), which on using eorem 1 gives and similarly, Journal of Mathematics which implies that where R(c + 2) > 0, R(P) > 0, and ξ ∈ U.
Proof. Since T c,P q (τ), then under the hypothesis of eorem 2, we have where A k,q (P, c, τ) and C are given by (34) and (61), respectively, and this last inequality, when combined with the following inequality (which is conveniently obtained from (20)), and using (63), we have |ξ| − B(c, P, m, q)C|ξ| m+1 ≤ |f(ξ)| ≤ |ξ| + B(c, P, m, q)C|ξ| m+1 ,  (20) be in the class T c,P q (τ), then for all ξ ∈ U, R(c + λ + 2) > 0, R(c + 2) > 0, e fractional q-calculus operators presented in Section 2 may be used to explore numerous different multivalent (or meromorphic) analytic function subclass and geometric characteristics which includes coefficient estimates, distortion bounds, radii of starlikeness, convexity, and so forth. e concept of fractional q-calculus can also be used to again with considerations.

Data Availability
No data were used to support this study.

Conflicts of Interest
ere are no conflicts of interest regarding the publication of this article.