Abstract
This article introduces a generalization for the Srivastava-Owa fractional operators in the unit disk. Conditions are given for the fractional integral operator to be bounded in Bergman space. Some properties for the above operator are also provided. Moreover, applications of these operators are posed in the geometric functions theory and fractional differential equations.
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1 Introduction
Recently, the theory of fractional calculus has found interesting applications in the theory of analytic functions. The classical definitions of fractional operators and their generalizations have fruitfully been applied in obtaining, for example, the characterization properties, coefficient estimates [1], distortion inequalities [2] and convolution structures for various subclasses of analytic functions and the works in the research monographs. In [3], Srivastava and Owa gave definitions for fractional operators (derivative and integral) in the complex z-plane C as follows:
Definition 1.1. The fractional derivative of order α is defined, for a function f(z), by
where the function f(z) is analytic in simply-connected region of the complex z-plane C containing the origin, and the multiplicity of (z - ζ)-αis removed by requiring log(z - ζ) to be real when (z - ζ) > 0.
Definition 1.2. The fractional integral of order α is defined, for a function f(z), by
where the function f(z) is analytic in simply-connected region of the complex z-plane (C) containing the origin, and the multiplicity of (z - ζ)α-1is removed by requiring log(z - ζ) to be real when (z - ζ ) > 0.
Remark 1.1. From Definitions 1.1 and 1.2, we have and . Moreover,
and
Further properties of these operators can be found in [4, 5].
2 Generalized integral operator
For 0 < p < 1, the Bergman space is the set of functions f analytic in the unit disk U := {z : z ∈ C; |z| < 1} with , where the norm is defined by
and dA is denoted Lebesgue area measure.
To derive a formula for the generalized fractional integral, consider for natural n ∈ N = {1, 2,...} and real μ, the n-fold integral of the form
By employing the Dirichlet technique yields
Repeating the above step n - 1 times we have
which implies the fractional operator type
where a and μ ≠ -1 are real numbers and the function f(z) is analytic in simply-connected region of the complex z-plane C containing the origin, and the multiplicity of (z μ+1- ζ μ+1)-αis removed by requiring log(z μ+1- ζ μ+1) to be real when (z μ+1- ζ μ+1) > 0. When μ = 0, we arrive at the standard Srivastava-Owa fractional integral, which is used to define the Srivastava-Owa fractional derivatives.
Theorem 2.1. Let α > 0,0 < p < ∞ and μ ∈ R. Then, the operator is bounded in and
where
Proof. Assume that . Then, we have
Let , then we obtain
This completes the proof.
Next, we give semigroup properties of the integral operator.
Theorem 2.2. Let f be analytic in the unit disk. Then, operator (2) satisfies
Proof. For function f by using Dirichlet technique yields
Let , we pose
By (4) and (5), we obtain
Example 2.1. We find the generalized integral of the function f(z) =z ν,ν ∈ ℝ. Let then
where B is the Beta function. When μ = 0, we obtain (see Remark 1.1).
In the next section, we will define generalized fractional derivatives for an arbitrary order. Some of its properties are discussed. Furthermore, applications involving this operator are illustrated.
3 Generalized differential operator
Corresponding to the generalized fractional integrals (2), we define the generalized differential operator.
Definition 3.1. The generalized fractional derivative of order α is defined, for a function f(z), by
where the function f(z) is analytic in simply-connected region of the complex z-plane C containing the origin, and the multiplicity of (z μ+1- ζ μ+1)-αis removed by requiring log(z μ+1- ζ μ+1) to be real when (z μ+1- ζ μ+1) > 0.
Example 3.1. We find the generalized derivative of the function f(z) = z ν, ν ∈ R. In the same manner of Example 2.1, we let then we have
When μ = 0, we obtain (see Remark 1.1).
Next, we proceed to prove some relations of the generalized operators and for analytic functions of the form
By employing Theorem 2.2 and Example 2.1, we have the following proposition:
Proposition 3.1. Let f be analytic in U of the form (8). Then,
4 Applications
In this section, we discuss some applications of the generalized operators (2) and (7) in geometric function theory and fractional differential equations.
4.1 Distortion inequalities involving fractional derivatives
Let denote the class of functions f(z) normalized by
Also, let and denote the subclasses of consisting of functions which are, respectively, univalent and convex in U. It is well known that if the function f(z) given by (9) is in the class , then
Equality holds for the Koebe function
Moreover, if the function f(z) given by (9) is in the class , then
Equality holds for the function
In our present investigation, we shall also make use of the Fox-Wright generalization q Ψ p [z] of the hyperge-ometric q F p function defined by [6]
where A j > 0 for all j = 1,..., q, B j > 0 for all j = 1,..., p and for suitable values |z| < 1.
Theorem 4.1. Let . Then,
where the equality holds true for the Koebe function.
Proof. Suppose that the function is given by (9). Then, by using Example 3.1, we obtain
Thus,
In the same manner of Theorem 4.1, we have a distortion inequality involving the Fox-Wright function, which is given by the following:
Theorem 4.2. Let . Then,
where the equality holds true for the Koebe function.
Theorem 4.3. Let . Then,
where the equality holds true for the Koebe function.
Proof. Suppose that the function . is given by (9). Then, we pose
4.2 Fractional differential equations
In this section, we focus our attention on the fractional differential equation of the form
subject to the initial condition u(0) = 0, where u : U → C is an analytic function for all z ∈ U, and f : U × C → C is an analytic function in z ∈ U. Let represent complex Banach space of analytic functions in the unit disk.
Theorem 4.4. (Existence) Let the function f : U × C → C be analytic such that ||f|| ≤ M; M ≥ 0. Then, there exists a function u : U → C solving the problem (15).
Proof. Define the set , and the operator P : S → S by
First, we show that P is bounded operator:
that is . We proceed to prove that P : S → S is continuous operator. Since f is continuous function on U × S, then it is uniformly continuous on a compact set , where
Hence, given ϵ > 0, ∃δ > 0 such that for all u,v ∈ S we have
Thus, P is a continuous mapping on S. Now, we show that P is an equicontinuous mapping on S. For such that z 1 ≠ z 2, then for all u ∈ S we obtain
which is independent on u. Hence, P is an equicontinuous mapping on S. The Arzela-Ascoli theorem yields that every sequence of functions from P(S) has got a uniformly convergent subsequence, and therefore P(S) is relatively compact. Schauder's fixed point theorem asserts that P has a fixed point. By construction, a fixed point of P is a solution of the initial value problem (15).
Theorem 4.5. (Uniqueness) Let the function f be bounded and fulfill a Lipschitz condition with respect to the second variable: i.e.,
for some L > 0 independent of u,v and z. If , then there exists a unique function u : U → C solving the initial value problem (15).
Proof. We need only to prove that the operator P in Equation 3 has a unique fixed point.
Then, for all u,v, we obtain
Thus, the operator P is a contraction mapping then in view of Banach fixed point theorem, P has a unique fixed point which corresponds to the solution of the initial value problem (15).
5 Conclusion
From above, we made a generalization to one of the most important differential and integral operators (Srivastava-Owa operators) of arbitrary order in the unit disk. We found that the generalize integral operator satisfying the semi-group property. Furthermore, their applications appeared in the theory of geometric functions and fractional differential equations by establishing the sufficient conditions for the existence and uniqueness of Cauchy problem in the unit disk.
References
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The author thankful to the anonymous referee for his/her helpful suggestions for the improvement of this article.
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Ibrahim, R.W. On generalized Srivastava-Owa fractional operators in the unit disk. Adv Differ Equ 2011, 55 (2011). https://doi.org/10.1186/1687-1847-2011-55
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DOI: https://doi.org/10.1186/1687-1847-2011-55