The fractional integrodifferential operator and its univalence and boundedness features according to Pre-Schwarzian derivative structure

Complex-valued regular functions that are normalized in the open unit disk are vastly studied. The current study introduces a new fractional integrodifferential (non-linear) operator. Based on the pre-Schwarzian derivative, certain appropriate stipulations on the parameters included in this constructed operator to be univalent and bounded are investigated and determined


Introduction
Over the past few decades, the theme of fractional calculus (FRC) has been widely, studied principally due to its significant implementations in mathematics and other fields related to it.Specifically, analysis (complex and mathematical) considerably evolved from FRC, which includes the incipient notions and analysis' techniques [12], [13].The foundations of the FRC theme were posed by Leibnitz in 1695.Its main concern is to study the extension of order derivatives and integrals to fractional derivatives and fractional integrals.
FRC has recently attracted the concern of the analytical community to the geometric complex function theory (GCFT).This great interest is due to its use as a valuable tool in researching a variety of operators with successful implementations in GCFT.In reality, Srivastava and Owa have introduced lots of contributions to developing on the theory FRC in the complex unit disk, such as [18], [21], and [22].Since then, many authors have provided to this area.For instance, Amsheri and Zharkova [3], Farzana et al [8], Ghanim and Al-Janaby [9], [10], [11]. Let indicate the unit disc in C "complex plane".The class of all regular functions in ∆ is denoted by E( ) ∆ .Consider : ' the subclass of E( ) ∆ that includes normalized regular functions f of the formulas: Besides, let / ' identify the subclass of : ' involving univalent functions.Denote by C V and S * consecutively the subclasses of : ' involving starlike and convex functions.Functions f ∈ C V map ∆ onto convex domain, while f ∈ S * whenever f ( ) ∆ starlike domains with respect to the origin.Analytically, , we denote the subclasses of : ' consisting of convex functions and starlike functions of order ε by C V ε and S ε * , consecutively.These geometric subclasses achieve the following series of proper inclusions: C S V H H * / ' .Moreover, they are regularly acquainted by and Obviously, for ε = 0 , then C V ε and S ε * coincide with C V and S * , sequentially, [14].The significant connection between the subclasses C V ε and S ε * , called "Alexander-sort", which achieves f C V H if and only if zf c S H * , [1].The function f is obtained from g as follows: where g is closer to similarities than g, see [2] and [6].Interesting investigations continued into certain subclasses of : ' .This includes the class L ∆ consisting of locally univalent functions in ∆ , namely, is a vector space over C in the sense of the Hornich operations [15].For f L ' , the term "pre-Schwarzian derivative" P f z ( ) is the logarithmic derivative of f and given by the following formula: Furthermore, the norm of P f is stated as: see [4].The pre-Schwarzian derivative is important in the theory of Teichmüller spaces and has a number of implementations in the theory of locally univalent functions.This norm is widely utilized in the study of geometric features of such functions.Specifically, it can be utilized to acquire either necessary or suitable stipulations for the global univalence or to gain certain geometric stipulations on the range of the function.Also, it is well known that any univalent regular transformation in ∆ [4].Recently, several authors have analyzed and introduced the norm estimates for typical subclasses of univalent functions, [19] and [20].
The following theorem has salutary tools discussing the major outcomes:

The constants are sharp
The first part is posed by Becker ([4], p. 36, corollary 4.1), while the sharpness of the constant 1 is due to Becker and Pommerenke [5].The second part is offered by Kim and Sugawa.[16].
On the other hand, the following fractional calculus (differential and integral) operators in the sense of Srivastava and Owa operators [22]: for function f, the fractional derivative of order α is given by where function f is regular in a simply connected region of C , including the origin, and the multiplicity of ( ) z W D is extracted by demanding log( ) z W to be real when 0 < ( ) z W . Whilst the fractional integral of order α is given by where function f is regular in a simply connected region of C , including the origin, and the multiplicity of ( ) 1 z W D is removed by demanding log( ) z W to be real when 0 < ( ) z W .The following lemma gives analogous formulations to the above concepts of fractional operators: In this paper, a new fractional integrodifferential operator is provided.Related to the pre-Schwarzian derivative, certain appropriate stipulations on the parameters included in this constructed operator to be univalent and bounded are considered and discussed.

Fractional Integrodifferential Operator S µ µ ( ) z
This section presents, for f : ' , a new fractional integrodifferential (non-linear) operator based on fractional binomial expansion and fractional differential formal operator in the sense of Srivastava and Owa operators.
In terms of fractional binomial expansion, 0 <1 d D , we consider a new fractional regular function for f : ' as: where G D N ( ) is the coefficients depending on a κ of f.In view of the fractional differential formula in the sense of Srivastava and Owa operators given by Theorem 3, the new fractional function ϑ f (10) yields the following fractional differential operator: = ( Then, we impose the following new modified classes of convex and starlike functions correlated with the fractional operator (11): Let C V H D ( ) denote the class of functions f : ' which achieve the condition where Also, let S H D * ( ) be the class of functions f : ' achieving where Therefore, based on the study in [23] and the fractional operator given by ( 11), let's introduce a fractional integrodifferential (non-linear) operator S P P ( ) : z : : ' ' o as: where f ι , V L : ' , 0 d K L , ι for L P = 1,2,..., and P ».

Univalence and boundedness of S µ µ ( ) z
This section investigates certain appropriate stipulations for univalence and boundedness of new fractional integrodifferential operator S µ ( ) z given by ( 14).An implementation of Theorem 1.1 and 1.2 to the new fractional operator ( 14) acquires the following major outcome.
then S µ ( ) z is bounded.
Proof.In light of ( 7) and ( 14), we acquire By an application of Theorem 1.2, this gains From inequality (19), it leads to Hence, by the first part of Theorem 1.1, S µ ( ) z is univalent.Furthermore, from condition (20), we deduce Thus, in view of the second part of Theorem 1 S µ ( ) z is bounded.

S
. In view of the Alexander-sort relation, it follows that there is D z ), it follows that D z f z