Analytic Normalized Solutions of 2D Fractional Saint-Venant Equations of a Complex Variable

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.


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][4][5][6][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.
Definition 5. The Fox-Wright function p W q ðzÞ (the extension function of hypergeometric function) is formulated by And it normalized by Note that the series is converged when Moreover, it converges for all finite values z to the entire function provided ⊳< − 1: In addition, at the boundary |z | = 1, 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 θ > 0 as follows: Based on this generalization, the authors in [24] introduced the following lemma. where is the delta-neutral H function and Π indicates the Fox-Wright coefficients.

Journal of Function Spaces
Then, a computation implies that Now, for the derivative, we have This completes the proof.
More integral inequality results will consider in the following theorem.

Journal of Function Spaces
LðzÞ ∈ H ½0, 1 with Proof. Let ϕ be convex univalent in ∪: Then, in view of Proposition 7, we have Consequently, by assuming r ⟶ 1, we obtain By 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. ?
GðzÞ ∈ H ½0, 1 with where g ∈ H ½1, 1, g ≠ 0: Proof. Let ϕ be convex univalent in ∪: Then, in view of Proposition 9, we have Consequently, by assuming r ⟶ 1, we have By 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. ?