Boundary fractional differential equation in a complex domain
© Ibrahim and Jahangiri; licensee Springer. 2014
Received: 2 December 2013
Accepted: 11 March 2014
Published: 24 March 2014
We discuss univalent solutions of boundary fractional differential equations in a complex domain. The fractional operators are taken in the sense of the Srivastava-Owa calculus in the unit disk. The existence of subsolutions and supersolutions (maximal and minimal) is established. The existence of a unique univalent solution is imposed. Applications are constructed by making use of a transformation formula for fractional derivatives as well as generalized fractional derivatives.
Fractional calculus is the most significant branch of mathematical analysis that transacts with the potential of covering real number powers or complex number powers of the differentiation operator . This concept was harnessed in geometric function theory (GFT). It was applied to derive different types of differential and integral operators mapping the class of univalent functions and its subclasses into themselves. Hohlov [1, 2] imposed sufficient conditions that guaranteed such mappings for the operators defined by means of the Hadamard product (or convolution) with Gauss hypergeometric functions. This was further extended by Kiryakova and Saigo  and Kiryakova [4, 5] to the operators of the generalized fractional calculus (GFC) consisting of product functions of the Gaussian function, generalized hypergeometric functions, G-functions, Wright functions and Fox-Wright generalized functions as well as rendering integral representations by means of Fox H-functions and the Meijer G-function. These techniques can be used to display sufficient conditions that guarantee mapping of univalent functions (or, respectively, of convex functions) into univalent functions. For example, for the case of Dziok-Srivastava operator see , and for an extension to the Wright functions see  which is concerned with the Srivastava-Wright operator. With the help of operators introduced in  and  one can establish univalence criteria for a large number of operators in GFT and GFC and for many of their special cases such as operators of the classical fractional calculus. Srivastava and Owa  generalized the definitions of fractional operators as follows.
where the multiplicity of is removed by requiring to be real when . Moreover, when , we have .
where the multiplicity of is removed by requiring to be real when .
where is the open unit disk and f is analytic in U satisfying the Riemann mapping conditions. The existence of subsolutions and supersolutions (minimal and maximal) is established. The existence of a unique univalent solution is introduced. Applications are also constructed by making use of some transformation formula for fractional derivatives. Equation (1) is a generalization of Beurling problem.
Let be the set of analytic functions f on the unit disk U normalized by and . And let be the subset of normalized by and . We denote by the set of all univalent functions .
Then every function is a subsolution for Ψ. If f is univalent, then it is called a univalent subsolution for Ψ.
Then every function is a supersolution for Ψ. If f is univalent, then it is called a univalent supersolution for Ψ.
Any subsolution for Ψ has a (Lipschitz) continuous extension to the closed unit disk . The set is uniformly bounded on and equicontinuous on .
- 2.A function with a continuous extension to U is a subsolution for Ψ if and only if
where is the Poisson kernel.
If a sequence of subsolutions from converges locally uniformly in U to a function (algebra of a holomorphic function that satisfies , ), then .
Let and let be a positive, continuous and bounded function with . Then, for all sufficiently close to 1, the function , , is a subsolution for .
Lemma 2.2 [, Lemma 2.7, Lemma 3.7]
Let Ψ be a positive, continuous and bounded function on ℂ. Assume that , are two subsolutions for Ψ (univalent supersolutions for Ψ). Then the upper of and is also a subsolution for Ψ (a univalent supersolution for Ψ).
Lemma 2.3 If is a solution to problem (1), then f has 0 as its unique critical point.
Lemma 2.3 is a generalization of the result found in . Thence, we cancel the proof.
Next result shows some properties of the set , which basically is a generalization of [, Lemma 3.3]. So we skip the proof.
Any univalent supersolution g for Ψ satisfies .
- 2.A bounded univalent function belongs to if and only if
where is the Poisson kernel.
If a uniformly bounded sequence of univalent supersolutions for Ψ converges locally uniformly in U, then the limit function f is again a univalent supersolution for Ψ.
Let be a positive, continuous and bounded function with , and let g be a univalent supersolution for Ψ. If g is bounded, then, for all sufficiently close to 1, the function , , is a univalent supersolution for Φ.
3 Main results
Our aim is to establish the largest univalent solution and the smallest univalent solution . We are able to state and prove the following theorem.
Theorem 3.1 Let Ψ be a positive continuous function on ℂ. Then there exists a unique univalent function (algebra unit disk), , such that and . Furthermore, the maximal subsolution is a solution.
This implies that with .
is identically equal to zero. Hence is a solution. This completes the proof. □
where f is analytic in U satisfying the Riemann mapping conditions.
As a consequence of our theorem, we have the following.
Corollary 3.1 Let Ψ be a positive continuous function on ℂ. Then there exists a univalent function satisfying and such that is a solution to (1).
Corollary 3.2 Let D be a simply-connected region in ℂ and be an analytic function on the open unit disk. If Ψ is a positive convex function, then there exists a unique univalent function satisfying and such that is a solution to (1).
Then there exists a univalent function satisfying and such that is a solution to (1).
Now, in a manner similar to Theorem 3.1, we have the following theorem.
which includes . Now the proof is complete by proceeding with a similar manner to that of the first part of the proof of Theorem 3.1. □
As a consequence of Theorem 3.2 above, we have the following.
Corollary 3.4 Let Ψ be a positive continuous function on ℂ. Then there exists a univalent function satisfying and , where solves problem (2).
Consequently, for the class consisting of analytic functions that are univalent in U, we have the following upper bound of the operator .
Finally, by letting the Koebe function for in (3), we can show that the result is sharp. Hence the proof. □
Moreover, we can prove the following theorem.
where F is a hypergeometric function.
We obtain the following two corollaries by making use the above operator and, respectively, letting and .
In the following example we demonstrate that, in view of Theorem 3.1, the above boundary problems have univalent solutions in the unit disk with .
It is clear that , as well as , which satisfies , .
By using the same technique as in the first part of Theorem 3.1, together with Lemma 2.2 and Lemma 2.4, we conclude the following.
We call the function of Theorem 3.5 the minimal univalent supersolution for Ψ.
This work is supported by University of Malaya High Impact Research Grant no vote UM.C/625/HIR/MOHE/SC/13/2 from the Ministry of Higher Education Malaysia. The authors also would like to thank the referees for giving some suggestions for improving the work.
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