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Abstract
A generalized differential operator utilizing Raina's function is constructed in light of a certain type of fractional calculus. We next use the generalized operators to build a formula for analytic functions of type normalized. Our method is based on the concepts of subordination and superordination. As an application, a class of differential equations involving the suggested operator is studied. As seen, the solution is provided by a certain hypergeometric function. We also create a fractional coefficient differential operator. Its geometric and analytic features are discussed. Finally, we use the Jackson's calculus to expand the Raina's differential operator and investigate its properties in relation to geometric function theory.
1. Introduction
Fractional calculus has grown in popularity in recent years, thanks to its applications in science and engineering. Fifty-first-order differential equations are used to model almost all nonlinear physical processes. In terms of the Mittag–Leffler function and its extensions, all classes of fractional differential equations have solutions in terms of this function (the Queen Function of Fractional Calculus) (see [Citation1–3]).
Basic power sums and polynomials, particularly the Mittag–Leffler function and its generalizations (Raina's function), as well as polynomials and their implications, are recognized to have extensive applications in various areas of number theory, such as the theory of partitions. Vector calculus, statistical studies, particle physics, optics, fluid studies, mechanical studies, quantum theory and applications, thermal study, and measurements all benefit from these functions (see [Citation4–12]). This function has been investigated in different types of inequalities and convex inequalities. Shu-Bo Chen et al. [Citation13] presented an integral formula inequality containing the Raina's function. Chu et al. [Citation14] generalized harmonically ψ-convex with respect to Raina's function on fractal set. Rashid et al. [Citation15] extended the Mittag–Leffler kernel. Mohammed et al. [Citation16–18] introduced various studies on the generalized Mittag–Leffler kernel.
In this study, we look at how Raina's function
may be used to extend a differential operator in the open unit disk. The fractional differential operator is employed to explain a variety of innovative normalized analytic functions. Therefore, we utilize the convolution product between the normalized Raina's function and analytic function satisfying the normalization equality
. To investigate a collection of differential inequalities, the concept of differential subordination and superordination is employed. Furthermore, we investigate the geometric behaviour of the diffusive wave differential equation, a family of analytic functions. The novel convolution linear operator is used for a variety of applications.
2. Approaches
In this section, we'll go through the approaches we employed.
2.1. Geometric approaches
We'll go over some geometric function theory fundamentals covered in this book [Citation19]
Definition 2.1
Introduce the set , which indicates the open unit disk. The analytic functions
in
are subordinated
or
if for an analytic function
fulfilling
Definition 2.2
Define the subclass of analytic functions
denoting by Λ and satisfying
.
Moreover, two functions are convoluted (
) if they achieve the product [Citation20]
Definition 2.3
Related to this class, the class of starlike functions and the class
of convex functions. Moreover, the class
is a special class of analytic functions in
with positive real part in
and
.
2.2. Raina's function
Integrals and outcomes of many kinds of differential equations fall within the category of special functions. As a result, most integral sets contain descriptions of special functions, and these special functions entail the most fundamental integrals; at the very least, the integral representation of special functions. Because symmetries of differential equations are significant in both physics and mathematics, the theory of special functions is closely connected to several mathematical physics issues [Citation21]. To begin, we'll look at the Mittag–Leffler function, which is a well-known special function.
Definition 2.4
The power of the generalized Mittag–Leffler function is as follows: [Citation4]
where
is the gamma function and
is the Pochhammer operator. Obviously, we have [Citation10]
Continue by defining Raina's function.
Definition 2.5
The power series determines Raina's function as follows [Citation22]:
where
and
is a bounded sequence of positive real numbers.
Remark 2.6
If
then we have
;
If
then we obtain
;
If
then we receive the hypergeometric function
Utilizing the function , we the convolution operator, for
where
Now by using the Sàlàgean derivative [Citation23], we have
Clearly,
. As a result, the Raina fractional differential operator can be studied geometrically.
Remark 2.7
The linear operator
is a natural transform of the analytic function
(
). The Raina's summation, which is a generalization of the Mittag–Leffler summation, is the name for this function
And the Borel's summation
In the geometric function theory, the operator
is a generalization of the well-known linear Carlson–Shaffer operator [Citation24], when
and
such that
When
for all
we obtain the well known the Sàlàgean differential operator [Citation23]
2.3. Preparatory
The conclusions of this investigation into the differential subordination theory are established using the following preliminaries:
Lemma 2.8
[Citation19]
Suppose that and
are convex univalent defining in
with
. In addition, for a constant
, the subordination
yields
Lemma 2.9
[Citation19]
Define the general class of analytic functions
where
and n is a positive integer. If
, then
Furthermore, if
and
then there occurs two positive numbers
and
satisfying the relation
implies
Lemma 2.10
see [Citation25]
Let , where υ is convex univalent in
and for
then
Lemma 2.11
see [Citation26]
Let , where ℘ is convex univalent in
and the functional
is univalent for
. Then
Lemma 2.12
[Citation27]
Assume that ℏ analytic in fulfilling
. Then the upper value of ℏ on the circle
at the point
is
where
represents the Jackson fractional derivative (or quantum fractional derivative).
3. Outcomes
In this study, we formulate the next class of normalized analytic functions and study its properties in view of the geometric function theory.
Definition 3.1
A function is called to be in the class
if it fulfils the inequality
(1)
(1) where ρ is convex univalent in
.
Obviously, the convex univalent function
is a member in the class
Consider the functional
as in the following structure:
(2)
(2) Based on Definition 3.1, we have the following inequality:
Our study is as follows:
3.1. Inequalities outcomes
We start with the next property of Raina' s operator.
Theorem 3.2
Let such that
Then the inequality is fully filled by the coefficient boundaries of
with the probability measure
:
Moreover, if
then
and
Proof.
Since
then
is a Carathéodory function in the open unit disk. Continuously, the Carathéodory positivist methodology brings that
where
is a probability measure. Additionally, if
then in virtue of [Citation20, Theorem 1.6] and for fixed number
we get
Moreover, we have from the proof of [Citation20, Theorem 1.6]
and that such that the range
i s contained in the interior of
. This yields
. Hence,
The next outcomes indicate the necessary and sufficient method for the functional sandwich theory.
Theorem 3.3
Let the following conditions hold:
(3)
(3) where
and convex in
. Additionally, assume that
is univalent in
such that
where
presents the set of all univalent analytic functions g with
and
(4)
(4) Then
and
is the best sub-dominant and
is the best dominant.
Proof.
Let
A computation implies
As a result, the following double inequality is obtained
As a conclusion, the desired result is yielded by Lemmas 2.10 and 2.11.
Theorem 3.4
Let
then
Proof.
A computation yields
According to Lemma 2.9 with
we get
3.2. Fractional differential equation
In this part, we continue our study using the convolution linear operator. We formulate the operator to present a generalized formula of the diffusive wave differential equation. When inertial acceleration is substantially lower than all other sources of acceleration, or when there is mostly sub-critical flow with low Froude values, the diffusive wave is viable.
In light of the suggested operator, we utilize the class to develop a class of fractional diffusive wave differential equations. We look at the upper bound of the diffusive wave equation. The formula is as follows:
(5)
(5) The solution to (Equation5
(5)
(5) ) is given by the following result.
Theorem 3.5
Let . Then (Equation5
(5)
(5) ) has a solution expressed by
(6)
(6) where
indicates the hypergeometric function.
Proof.
Assume that . Then it satisfies the differential equation
where φ is a Schwarz function with the property:
and
. Now, by using Schwarz lemma, equality
(see [Citation28, Theorem 5.34]) implies the differential equation
Rearrange the above equation, we have
Multiply the above equation by the functional
we have
The solution of the above first-order differential equation is
where
indicates the hypergeometric function. This completes the proof.
Example 3.6
Let where
. Then in view of Theorem 3.5, we get the solution
3.3. First order differential operator
In the next study, we employ the Raina's operator to define a new generalized differential operator.
Definition 3.7
For non-negative real numbers λ let be the integer part of λ. For
, and by employing the Raina's operator
we have the following extended linear differential operator:
(7)
(7) where for
,
where
and the functions
are analytic in
with
and
It is clear that, for constant coefficients, . For example
and
.
Clearly, if λ assumes only non-negative integer values, that is if , then we have the Sàlàgean differential operator [Citation23]. We also have the differential operator in [Citation29], which is based on the same assumptions. In this section, we examine the geometric properties of the complex conformable derivative (Equation7
(7)
(7) ) when applied to functions with a positive real portion.
Theorem 3.8
For a fixed number and
let
Then
Proof.
For and by Definition 3.7, we get
and
Obviously, we obtain
if and only if
Accordingly, if and only if
(8)
(8) The convexity of a function is obtained by combining the inequality Equation8
(8)
(8) with the idea of convex functions:
But all convex functions are starlike, then we obtain that
(9)
(9) The inequality Equation9
(9)
(9) occurs if and only if
and this ends the proof.
The main condition to put on the operator is computed by our second theorem, for the functional
to be of positive real part.
Theorem 3.9
For a positive number and
let
If
, then
Proof.
Applying the differential operator rule to
implies
(10)
(10) Dividing Equation Equation10
(10)
(10) by the term
and utilizing the relation
we get
The convexity of
it becomes
Hence, it yields that
This ends the proof.
3.4. Quantum starlike methodology
Quantum calculus (QC) is a novel field of mathematical analysis and its applications, with applications in physics and mathematics. Jackson [Citation30, Citation31] originally defined and enhanced the functions of q-differentiation and q-integration. The geometric function theory idea of q-calculus was later incorporated by Ismail et al. [Citation32]. QC is now being used by researchers to propose and build new Ma and Minda classes. Seoudy and Aouf [Citation33] suggested a quantum starlike function subclass based on q-derivatives. Recently, Zainab et al. [Citation34] employed a novel curve to create appropriate q-stralikeness criteria. Different types of q-stralik functions dominated by a 2D-Julia set were explored by Samir et al. [Citation35]. Furthermore, QC is used to generalize a variety of differential and integral operators [Citation36–42].
Definition 3.10
The Jackson derivative may be shown using the difference operator below.
(11)
(11) such that
The total of the numbers is also included in the Maclaurin's series representation.
(12)
(12) where
Note that
where ∁ is a constant. Then there's the multiplication rule, which is formulated by multiplying two numbers together
We then use the q-parametric Mandelbrot function to formulate our q-starlike class, linking it to the normalized function subclass in the process
(13)
(13) We aim to investigate the sufficient conditions on the two parameters ℓ and q to obtain the q-starlike function.
Theorem 3.11
Assume that with
and
(14)
(14) If for some positive constant j achieves the inequalities
(15)
(15) then for some
we have
(16)
(16)
Proof.
Formulate a function by
The condition (Equation14
(14)
(14) ) implies that
A computation gives
Our aim is to show that
where
satisfying
Consume not; if so, the preceding conclusion applies
Employing Jackson's derivative principles as well as the formula
and
we obtain
Consider the existence of a point
such that
and
We proceed to prove
utilizing Jack Lemma 2.12.
Letting we get
where
which contradicts (Equation15
(15)
(15) ). Hence, we obtain (Equation16
(16)
(16) ).
The following examples involve the Raina' s operator.
Example 3.12
For a positive number and
let
If
, then in view of Theorem 3.9, we have
That is
. Moreover, if
where q satisfies (Equation15
(15)
(15) ) then according to Theorem 3.11
If
then in view of Theorem 3.8, we get
That is
. In addition, if
where q satisfies (Equation15
(15)
(15) ) then by Theorem 3.11, we have
Let
where
. If
where q achieves the inequality (Equation15
(15)
(15) ) then in virtue of Theorem 3.11, we obtain
4. Conclusion
Raina's transformations in were generalized utilizing conformable calculus and Jackson calculus in the above investigation. The Raina's convolution operator is acted on the normalized subclass. As an application, we considered the proposed linear convolution operator in fractional differential equation, type wave equation. The solution of a certain type of diffusion differential equation, which is utilized as a case study, is determined by the hypergeometric function.
More investigation is presented by formulating the Raina's convolution operator in a conformable fractional calculus. We studied the main sufficient conditions to get stralike geometry of the operator (see Theorems 3.8 and 3.9).
Finally, the quantum calculus is utilized to recognize the q-starlike function together with the q-parametric Mandelbort function. As an application, we applied the result using the Raina's convolution operator (see Example 3.12).
Author contributions
All authors contributed equally and significantly to writing this article. All authors read and agreed to the published version of the manuscript.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Data availability statement
Data sharing not applicable to this article as no data-sets were generated or analysed during the current study.
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