Abstract
Saint-Venant equations describe the flow below a pressure surface in a fluid. We aim to generalize this class of equations using fractional calculus of a complex variable. We deal with a fractional integral operator type Prabhakar operator in the open unit disk. We formulate the extended operator in a linear convolution operator with a normalized function to study some important geometric behaviors. A class of integral inequalities is investigated involving special functions. The upper bound of the suggested operator is computed by using the Fox-Wright function, for a class of convex functions and univalent functions. Moreover, as an application, we determine the upper bound of the generalized fractional 2-dimensional Saint-Venant equations (2D-SVE) of diffusive wave including the difference of bed slope.
1. Introduction
Newly, fractional calculus has expanded considerable attention primarily appreciations to the growing occurrence of investigation mechanisms in the life sciences, allowing for simulations found by fractional operators [1] including differential and integral formulas. Further, the mathematical investigation of fractional calculus has advanced, chief to connections with other mathematical areas such as probability theory, mathematical physics [2], and mathematical biology [3–7] and the investigation of stochastic processes in real cases. In addition, it appears in studies of complex analysis. Now the literature, several different definitions of fractional integrals and derivatives are presented. Some of them such as the Riemann-Liouville integral, the Caputo, and the Riemann-Liouville differential operators are extensively employed in mathematics and physics and actually utilized in applied structures, modeling systems in real cases. While, in complex analysis, especially the theory of geometric functions, the researchers are focusing on Srivastava-Owa integral and differential operators [8], Tremblay differential operator, and the most recent fractional operator in [9, 10]. A new investigation of the complex ABC-fractional operator is presented to formulate different classes of analytic functions [11]. Some definitions such as the Hilfer and Prabhakar results [12] (differential and integral operators) are essentially the theme of mathematical study.
Our study is aimed to extend the Prabhakar operator [13] to the open unit disk utilizing the class of normalized analytic functions. We formulate this operates in a linear convolution operator to study some important geometric behaviors. A class of integral inequalities is investigated involving special functions. The upper bound of the suggested operator is computed by using the Fox-Wright function, for a class of convex functions and univalent functions, and other studies are illustrated in the sequel.
2. Complex Prabhakar Operator (CPO)
The Prabhakar integral operator is defined for analytic function by the formula [14–20] where, [15]
For example, let then in view of [21] Corollary 2.3, we have
The Prabhakar derivative can be computed by the formula [13]
To study the geometric indications of CPO, we introduce the following class of analytic function: a normalized analytic function achieving the power series
Two analytic functions are called convoluted, denoting by if and only if
Definition 1. Define a new function such thatNote that,where is a transcendental function. Utilizing the functional we define the modified complex linear Prabhakar operator
We have the following result.
Proposition 2. If then where indicates the convolution product.
Proof. Let then we havewhereHence,
We need the following concepts to study the geometrically.
Definition 3. A function is indicated to be starlike including the origin of the linear slice contains the origin to all further point of in . A univalent function (one-one) is indicated to be convex in if the linear slice relating every two points of lies completely in . We indicate these classes by and for starlike and convex consistently. Consider the class includes all mappings smooth in with a positive real part in realizing . Precisely, and . Regularly, for the starlikeness and for the convexity.
Two analytic functions in are called subordinated [22] denoting by or if there occurs an analytic function satisfying
Next, lemma can be located in [22], P138-140.
Lemma 4. LetThen,(a) when (b) when (c) when (d) when (e) when (f) when and the solution is sharp.
Definition 5. The Fox-Wright function (the extension function of hypergeometric function) is formulated byAnd it normalized byNote that the series is converged whenMoreover, it converges for all finite values to the entire function provided In addition, at the boundary it has the convergence value (see [23])The significance of the Fox-Wright function arises regularly from its part in fractional calculus (see [1]). Further fascinating applications correspondingly occur. Wright’s original attentiveness in this function was connected to the asymptotic theory of partitions [24]. The formula is generated in [23] by adding a positive parameter as follows:
Based on this generalization, the authors in [24] introduced the following lemma.
Lemma 6. Assume that and . Then,whereis the delta-neutral function and indicates the Fox-Wright coefficients.
Proposition 7. Let (convex in the open unit disk), then
Proof. Since then for each we have . Then, a computation implies thatNow, for the derivative, we haveThis completes the proof.
Remark 8. (i)It is clear that the above upper bound of converges atMoreover, the upper bound of converges, when(ii)For special case: by Proposition 7 and Lemma 6, we haveprovided that and
Proposition 9. Let be univalent in the open unit disk. Then,
Proof. Since is univalent, then for each we have Then a calculation indicates thatNow, we return to the upper bound of the derivativeThis completes the proof.
More integral inequality results will consider in the following theorem.
Theorem 10. Consider and the operator where If one of the following subordination inequalities holdthen and
Proof. Suppose that the operator achieves one of the subordination inequalities (a)-(h) then, in view of Proposition 2 and results in Lemma 4, we haveConsequently, we obtain the upper bound inequalityThe function achieves the real inequalityprovided that Moreover, we have superior inequalityHence, in view of [22] [Corollary 4.3a.2, P210], we conclude thatwhich yieldshence the proof.
Now, we investigate another integral inequality involving the operator where
Theorem 11. Consider and the operator where If one of the subordination inequalities in Theorem 10 holds, thenwhere
Proof. Suppose that the operator has one of the subordination inequalities (a)-(h) in Theorem 10 then, in view of Proposition 2, we havewhich implies thatAs in Theorem 10, the function admits the real inequalityprovided that Moreover, we have superior inequalityThus, in view of [22] [Theorem 4.3a, P207], we indicate that then andwhere , hence the proof.
In addition, we have the following result by replacing by :
Theorem 12. Consider and the operator where If one of the subordination inequalities in Theorem 10 holds, then
Proof. Suppose that the operator has one of the subordination inequalities (a)-(h) then, in view of Proposition 2 and results in [22], P138-140, we havewhich leads toThe function admits the following propertiesMoreover, we have superior inequalityThus, in view of [22]-[Corollary 4.3a.1, P208], andThis completes the proof.
Theorem 13. Consider that is convex univalent function in the open unit disk and the operator where Then,with withwhere
with
Proof. Let be convex univalent in Then, in view of Proposition 7, we haveConsequently, by assuming we obtainBy the proof of Theorem 10, we conclude that (A). Similarly, by using the proof in Theorems 11 and 12, we have (B) and (C), respectively. This ends the proof.
Theorem 14. Consider that is univalent function in the open unit disk and the operator where Then
with withwhere
with
Proof. Let be convex univalent in Then, in view of Proposition 9, we haveConsequently, by assuming we haveBy the proof of Theorem 10, we conclude that (A). Similarly, by using the proof in Theorems 11 and 12, we have (B) and (C), respectively. This ends the proof.
In the next result, we discuss the starlikeness of the operator
Theorem 15. Consider the operator (A)If where then (B)If and then(C)If then
Proof. For part (A), assume that thenConsequently, we haveThe value implies that Sincethen in view of [22]-[Theorem 5.5c, P296], we conclude that
For part (B), sincewhere the number 0.374 is a solution of the equation then by [22] [Theorem 5.5g, P299], we have Moreover, in terms of , we haveby [22] [Theorem 5.5d, P298] thenThe last part immediately comes from [22] [Corollary5.5.a,P294]. This ends the proof.
Theorem 16. Consider the operator (A)If the following inequality holdsthen(B)If the subordination occursthen(C)If the next relation existsthen
Proof. It is clear that the functionsatisfies and it is convex in the open unit disk. Consequently, it is starlike. By Proposition 2, the operator and hence that is Similarly for the functionThus, in view of [22]-[Corollary 3.1d.1, P76], we have the desire results.
For the last part (C), thus, in virtue of [22]-[Theorem 3.1d, P76], where we conclude the last subordination.
2.1. Fractional Saint-Venant Equations
By using the fractional calculus of the construction, we formulate the fractional 2D-Saint-Venant equations utilizing the functional convolution operator and
Example 1. We investigate the upper bound of the 2-dimensional Saint-Venant equations (2D-SVE) of diffusive wave (this equation has measured the level of the water). This equation simply presents the formulawhere is the height deviation of the horizontal pressure surface at two-dimensional position and represents the difference of bed slope. By using the convolution operator, we generalize 2D-SVE into the formMultiplying both sides of Eq. (76) by and letwe haveThus, in view of Theorem 16-(C), we conclude the upper solution of Eq. (78) is given by (see Figure 1, second row)Hence, we obtainwhere is a constant andindicates the exponential integral. Assuming thatwe getBy the convexity of (see [22]-P139), we confirm that the solution is normalized analytic convex in Note that the term is called the convective acceleration term. Figure 1 shows the behavior of solutions of 2D-SVE of diffusive wave.
(a) The first row represents the function and the second row indicates the solution (79) of the extended 2D-SVE
(b) The solution
3. Conclusion
From above, we have extended the Prabhakar operator in the open unit disk. We formulated it in a linear convolution operator with a normalized function. A class of integral inequalities is investigated involving special functions. The upper bound of the suggested operator is computed by using the Fox-Wright function, for a class of convex functions and univalent functions. Moreover, we applied the operator to generalize the 2D-SVE. A solution of the extended 2D-SVE is computed by using recent result (Theorem 16).
For future woks, one can consider extra studies in the geometric function theory by considering the operator in different classes of analytic functions, such as normalized functions, harmonic functions, and meromorphic functions.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors contributed equally and significantly to writing this article. All authors read and agreed to the published version of the manuscript.