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.
Remark 1.1 From the above two definitions, we observe that
In this paper, we study univalent solutions of boundary fractional differential equations in a complex domain. The fractional operators are considered in the sense of the Srivastava-Owa  differential operator
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.
In this work, we utilize the generalized sets
Lemma 2.1Let Ψ be a positive, continuous and bounded function on ℂ.
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 ofandis also a subsolution for Ψ (a univalent supersolution for Ψ).
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.
Lemma 2.4Let Ψ be a positive, continuous and bounded function on ℂ.
3. If a uniformly bounded sequence of univalent supersolutions for Ψ converges locally uniformly inU, then the limit functionfis again a univalent supersolution for Ψ.
4. Letbe a positive, continuous and bounded function with, and letgbe a univalent supersolution for Ψ. Ifgis bounded, then, for allsufficiently close to 1, the function, , is a univalent supersolution for Φ.
3 Main results
and for sufficiently small values of α, we have
we conclude that
Therefore, by virtue of the principle of subordination, this yields
where φ is the harmonic function on whose boundary values are logΨ (see [, p.402]). From the definition of Φ, we may conclude that is the maximal subsolution of Φ. Let be a solution for (1) having a unique critical point at 0 (Lemma 2.3). Obviously, and the two functions and are continuous on , harmonic on U and coincide on ∂U. Thus
on U and
In addition, we conclude that
Remark 3.1 Note that Theorem 3.1 can be introduced for the boundary problem
where f is analytic in U satisfying the Riemann mapping conditions.
As a consequence of our theorem, we have the following.
Corollary 3.2LetDbe a simply-connected region in ℂ andbe an analytic function on the open unit disk. If Ψ is a positive convex function, then there exists a unique univalent functionsatisfyingandsuch thatis a solution to (1).
Corollary 3.3Let Ψ be a positive continuous sublinear function on ℂ; i.e.,
Next, we discuss the boundary problem for some functions , where . From [, Theorem 1.3], for , we define the fractional transform
Now, in a manner similar to Theorem 3.1, we have the following theorem.
Theorem 3.2Consider the problem
As a consequence of Theorem 3.2 above, we have the following.
Tremblay  studied a fractional calculus operator defined in terms of the Riemann-Liouville fractional differential operator. We extend this operator in the complex plane to involve as follows:
whereFis the hypergeometric function. The equality holds true for the Koebe function
Moreover, we can prove the following theorem.
whereFis a hypergeometric function.
Corollary 3.5Consider the problem
Corollary 3.6Consider the problem
Example 3.1 A computation implies
Thus, we have the boundary problems
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.
The authors declare that they have no competing interests.
Both authors jointly worked on deriving the results and approved the final manuscript.
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.
Kiryakova, V: Criteria for univalence of the Dziok-Srivastava and the Srivastava-Wright operators in the class A. Appl. Math. Comput.. 218, 883–892 (2011). Publisher Full Text
Srivastava, HM: Some Fox-Wright generalized hypergeometric functions and associated families of convolution operators. Appl. Anal. Discrete Math.. 1, 56–71 (2007). Publisher Full Text
Srivastava, HM, Darus, M, Ibrahim, RW: Classes of analytic functions with fractional powers defined by means of a certain linear operator. Integral Transforms Spec. Funct.. 22, 17–28 (2011). Publisher Full Text
Sokół, J, Piejko, K: On the Dziok-Srivastava operator under multivalent analytic functions. Appl. Math. Comput.. 177, 839–843 (2006). Publisher Full Text
Piejko, K, Sokół, J: Subclasses of meromorphic functions associated with the Cho-Kwon-Srivastava operator. J. Math. Anal. Appl.. 337, 1261–1266 (2008). Publisher Full Text
Sokół, J: On some applications of the Dziok-Srivastava operator. Appl. Math. Comput.. 201, 774–780 (2008). Publisher Full Text
Sivasubramanian, S, Sokół, J: Hypergeometric transforms in certain classes of analytic functions. Math. Comput. Model.. 54, 3076–3082 (2011). Publisher Full Text
Hussain, S, Sokół, J: On a class of analytic functions related to conic domains and associated with Carlson-Shaffer operator. Acta Math. Sci. Ser. B. 32, 1399–1407 (2012). Publisher Full Text
Černe, M, Zajec, M: Boundary differential relations for holomorphic functions on the disc. Proc. Am. Math. Soc.. 139, 473–484 (2011). Publisher Full Text
Gaboury, S, Tremblay, R: A note on some new series of special functions. Integral Transforms Spec. Funct. (2013). Publisher Full Text