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

Abstract

Through applying the Kober fractional -calculus apprehension, we preliminary implant and introduce new types of univalent analytical functions with a -differintegral operator in the open disk . The coefficient inequality and distortion theorems are among the results examined with these forms of functions. Specific cases are responded and addressed immediately. The findings include an expansion of the numerous established results in the -theory of analytical functions.

1. Introduction and Preliminary

The -analysis theory has been applied in recent times in several fields of science and engineering. The fractional -calculus is indeed an analog of the conventional fractional calculus in -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 -difference operators. Furthermore, Srivastava [3] investigated the excellent work with -calculus and fractional -calculus operators, which is quite valuable for academics working on these issues. The applications of fractional -calculus operators have been investigated by Purohit and Raina [4] to describe several new classes of analytic functions in open disk . 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 . Several others have also released new classes of analytical functions with the resources of -calculus operators. For any more inquiries on the analytic functions classes, we refer to [1, 2, 8–13] for functions described by applying -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 -calculus. The 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 -calculus which are relevant to grasp the object of the study.

For each complex number , the -shifted factorials are delimited byand with regard to the basic analog of the gamma function,in which the -gamma function is set by (see [14])

The recurrence relationship specified by Gaspar and Rahman [15] for the -gamma function is

If , then equation (1) shall continue to play a role as an infinite product of convergence:and we have

The -binomial expansion is now as follows:

[15] accounts for Jackson’s -integral and -derivative of a function , which are described on a subset of , aswith

2. The Fractional -Calculus Operators

Purohit and Raina [4] described the fractional -integral operator of function given bywhere is the order of integral and is an analytic function in , and (7) the be expressed aswhere is special case of basic hypergeometric series for is single valued for and  (see [15]).

Purohit and Raina [4] defined the fractional -derivative operator of a function bywhere and is suitably constrained with .

The Kober fractional -integral operator for a real valued function is determined by Garg and Chanchalani [16] aswhere being real or complex and is an absolute order of integration with . For , operator (13) is reduced to Kober operator as defined in [17]. For , this operator is converted to Riemann–Liouville fractional -integral operator with a power weight function as .

The Kober fractional -derivative operator for a real valued function is detailed by Garg and Chanchalani [16] aswhere is order of derivative with and . For , operator (14) is reduced to Kober operator as defined in [17].

We are now defining -calculus operators with a view to applying these operators to the geometric function theory of complex analysis.

Definition 1. Kober fractional -integral operator:
For the function , the Kober fractional -integral operator is demarcated bywhere is the real or complex, is an absolute order of integration with , and the -binomial is expressed as in (11).
For , operator (15) is reduced to Kober integral operator as defined in [17].

Definition 2. Kober fractional -derivative operator:
The Kober fractional -derivative operator for the function is demarcated bywhere is the order of derivative with and . For , operator (16) is reduced to Kober derivative operator as defined in [17].
Under Kober -integral and -derivative operators fixed by (15) and (16), we offer the following image formulae for function .

Remark 1. If , , and , then

Remark 2. If , , and , then

3. New Classes of Functions

Let represent the function class of the formwhich are analytic and univalent in open unit disk . Above, let highlight the subclass of imposing of analytical and univalent functions articulated in the form

For the dedication of this work, we describe a fractional -differintegral operator for a function of the form (20) bywhereand , ,  ,  ,  , and represent a fractional -derivative of of order . We announce here the alike classes of functions connecting operator (21):where .

And

The subsequent coefficient bounds for functions of the form (20) that belong to the classes and are now obtained (interpreted above).

Theorem 1. A function defined by (20) is connected to the class if and only ifwhereThe result is sharp.

Proof. Let us consider that inequality (25) holds, and for , we haveand by our assumption, this indicates that .
For the proof of converse part, suppose that , and then it follows thatwhich implies thatSince for any , therefore on choosing values of on the real axis so that is real and allowing all through real values, we obtain from above inequalitywhich implies thatwhich is desired result. Here, we notice that assumption (25) of Theorem 1 is sharp and the external function is assumed bywhere is defined in (26).

Theorem 2. A function defined by (20) is connected to the class if and only ifwhereThe accomplishment is sharp.

Proof. To prove above theorem, we address the elementary assertion thatNow,whereIn (35), it then suffices to show thatNow,This accomplishes the proof of theorem.
We accommodate that answer (33) is sharp. The external function is assumed bywhere is given by (34).

4. Distortion Theorems

Theorem 3. Suppose that the function is defined by (20) in the class , thenwhereFurthermore,where .

Proof. Since , then in interpretation of Theorem 1, we first show that the functionis an increasing function of for and .
It follows thatTaking , thenThe function is an increasing function of if , and this giveswhich impliesThis inequality abides for .
Thus, is an increasing function of k for and .
Now, (25) gives the alike inequality:which implies thatwhere is defined in (42), and this last inequality is in the conjunction with the alike inequality (easily obtained from (20)):and using (50), we havewhich is result (41) of Theorem 3.
Now, on using (21), we observe that for functions of form (20),which on using Theorem 1 givesand similarly,which implies that

Corollary 1. Let the function detailed in (20) be in the class , thenwhere , and .

Corollary 2. Let the function detailed in (20) be in the class , thenwhere , and .

Theorem 4. Suppose that the function detailed in (20) be in the class , then for ,alsowhereand and are detailed in (34) and (42), respectively.

Proof. Since , then under the hypothesis of Theorem 2, we havewhich implies thatwhere 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 havewhich is result (59) of Theorem 4.
Now, from (21), we obtainon using (63), this implies thatsimilarly, we haveand on combining above two results, we have

Corollary 3. Let the function detailed in (20) be in the class , then for all , ,

The fractional -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. The concept of fractional -calculus can also be used to again with considerations.

Data Availability

No data were used to support this study.

Conflicts of Interest

There are no conflicts of interest regarding the publication of this article.