# Some properties for integro-differential operator defined by a fractional formal

- Zainab E. Abdulnaby
^{1}, - Rabha W. Ibrahim
^{2}and - Adem Kılıçman
^{3}Email author

**Received: **9 March 2016

**Accepted: **12 June 2016

**Published: **27 June 2016

## Abstract

Recently, the study of the fractional formal (operators, polynomials and classes of special functions) has been increased. This study not only in mathematics but extended to another topics. In this effort, we investigate a generalized integro-differential operator \(\mathfrak {J}_{m}(z)\) defined by a fractional formal (fractional differential operator) and study some its geometric properties by employing it in new subclasses of analytic univalent functions.

### Keywords

Fractional calculus Unit disk Analytic functions Integral operator Univalent functions Fractional differential operator## Background

The subject of fractional calculus (integral and derivative of any arbitrary real or complex order) has acquired significant popularity and major attention from several authors in various science due mainly to its direct involvement in the problems of differential equations in mathematics, physics, engineering and others for example Baskonus and Bulut (2015), Yin et al. (2015) and Bulut (2016). The fractional calculus has gained an interesting area in mathematical research and generalization of the (derivative and integral) operators and its useful utility to express the mathematical problems which often leads to problems to be solved see Yao et al. (2015), Baskonus (2016) and Kumar et al. (2016). Specifically, it utilized to define new classes and generalized many geometric properties and inequalities in complex domain. In another words, these operators are play an important role in geometric function theory to define new generalized subclasses of analytic univalent and then study their properties. By using the technique of convolution or Hadamard product, Sălăgean (1981) defined the differential operator \(\mathcal {D}_{n}\) of the class of analytic functions and it is well known as S\(\check{a}\)l\(\check{a}\)gean operator. Followed by Al-Oboudi differential operator see Al-Oboudi (2004). Several authors have used the S\(\check{a}\)l\(\check{a}\)gean operator to define and consider the properties of certain known and new classes of analytic univalent functions. We refer here some of them in recent years. Najafzadeh (2010) investigated a new subclass of analytic univalent functions with negative and fixed finitely coefficient based on S\(\check{a}\)l\(\check{a}\)gean and Ruscheweyh differential operators. Aouf et al. (2012) gave some results for certain subclasses of analytic functions based on the definition for S\(\check{a}\)l\(\check{a}\)gean operator with varying arguments. El-Ashwah (2014) used S\(\check{a}\)l\(\check{a}\)gean operator to define a new subclass of analytic functions and derived some subordination results for this subclass in open unit disk. Breaz et al. (2008) investigated a new general integral operator for certain holomorphic functions based on the S\(\check{a}\)l\(\check{a}\)gean differential operator and studied some properties for this integral operator on some subclasses of univalent function. Also, Deniz et al. (2012) defined a new general integral operator by considering the Hadamard product and gave new sufficient conditions for this operator to be univalent in \(\mathbb {U}\). Breaz et al. (2014) defined two general integral operators \(F_{\lambda }(z)\) and \(G_{\lambda }(z)\) and investigated some geometric properties for these operators on subclasses of analytic function in open unit disk.

In this paper, we define a generalized mixed integro-differential operator \(\mathfrak {J}_{m}(z)\) based on the concept of Breaz integral operator as well as the fractional differential operator and study some their geometric properties on some new subclasses in open unit disk.

## Preliminaries

*f*of form (1) which are univalent in \(\mathbb {U}\). We denote by \(\mathcal {S}^{*}(\beta )\) and \(\mathcal {K}(\beta )\), \(0 \le \beta < 1\), the classes of starlike function and convex function in \(\mathbb {U}\), respectively. For \(f \in \mathcal {A}\), Esa et al. (2016a) introduced the following differential operator \(\mathcal {T}^{\alpha ,\delta }: \mathcal {A}\rightarrow \mathcal {A},\)

*f*, which satisfies the following

*f*which satisfies the following

*f*is said to be in the class \(\mathcal {K}^{k}\mathcal {L}(\rho ,\varphi )\), if

*positive*real numbers, \(j=\lbrace 1,2,\ldots ,m \rbrace\), we define the integral operator \(\mathfrak {J}_{m}(z):\mathcal {A}^{m} \rightarrow \mathcal {A}\) by

###
*Remark 1*

###
*Remark 2*

###
*Remark 3*

###
*Remark 4*

## Main results

We start our first result.

###
**Theorem 1**

*Let*\(\nu _{j},\, \beta _{j}\)

*be positive real numbers,*\(j=\lbrace 1,2,\ldots , m\rbrace\).

*If*\(f_{j} \in \mathcal {M}^{k}(\lambda , \psi _{j} ),\, \psi _{j} > 1\)

*and*\(g_{j} \in \mathcal {N}^{k}(\lambda , \eta _{j}), \,\eta _{j} > 1, j= \lbrace 1,2, \ldots , m \rbrace\),

*then the integral operator*\(\mathfrak {J}_{m} (z)\)

*given by*(9)

*is in the class*\(\mathcal {N}^{k} (\lambda ,\rho )\),

*where*

###
*Proof*

Let \(k=0, m=1\) in Theorem 1, we have

###
**Corollary 1**

*Let*\(\beta ,\, \nu\)

*be*positive

*real numbers*.

*If*\(f \in \mathcal {M}(\psi ), \psi > 1\)

*and*\(g \in \mathcal {N}(\eta ), \eta > 1\),

*then*

*is in the class*\(\mathcal {N}(\rho )\),

*where*\(\rho = 1+ \beta (\psi -1)+\nu (\eta -1)\).

###
**Theorem 2**

*Let*\(\beta _{j},\,\nu _{j}\)

*be*positive

*real numbers*, \(j= \lbrace 1,2,\ldots ,m\rbrace\).

*We assume that*\(f_{j},\,j=\lbrace 1,2,\ldots ,m\rbrace\)

*are starlike functions by order*\(\frac{1}{\beta _{j}}\)

*and that is*\(f_{j} \in \mathcal {S}^{k}(\lambda ,\,\frac{1}{\beta _{j}})\)

*and*\(g_{j}\in \mathcal {K}^{k}\mathcal {L}(\rho _{j},\, \eta _{j})\), \(\rho _{j} \ge 1,\,0 \le \eta _{j} < 1, j=\lbrace 1,2,\ldots ,m \rbrace\).

*If*

*then*\(\mathfrak {J}_{m}(z)\)

*given by*(9)

*is in the class*\(\mathcal {K}^{k}(\lambda ,\omega )\)

*where*

###
*Proof*

By setting \(k=0\) and \(m=1\) in Theorem 2, we have the following result.

###
**Corollary 2**

*Let*\(\beta , \nu\)

*be*

*positive real numbers, We assume that*\(f \in \mathcal {S}^{*}(\frac{1}{\beta })\), \(g \in \mathcal {K}\mathcal {L}(\rho ,\, \eta ), \rho \ge 0\)

*and*\(0 \le \eta < 1\).

*If*

## Conclusion

In geometric function theory, we defined and studied a new integro -differential operator \(\mathfrak {J}_{m}(z)\), with a new classes of analytic and univalent functions. This operator is generalized and modified recent various fractional differential operators.

## Declarations

### Authors' contributions

The authors, ZA, RI and AK contributed equally to the writing of this paper. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to thank the referees for valuable suggestions and comments, which helped the authors to improve this article substantially.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Al-Oboudi FM (2004) On univalent functions defined by a generalized Sălăgean operator. Int J Math Math Sci 2004(27):1429–1436View ArticleGoogle Scholar
- Alexander JW (1915) Functions which map the interior of the unit circle upon simple regions. Ann Math 17(1):12–22View ArticleGoogle Scholar
- Aouf MK, El-Ashwah RM, Hassan AAM, Hassan AH (2012) On subordination results for certain new classes of analytic functions defined by using Salagean operator. Bull Math Anal Appl 4(1):239–246Google Scholar
- Baskonus HM (2016) Analytical and numerical methods for solving nonlinear partial differential equations—II. In: 1st international symposium on computational mathematics and engineering sciences, Errichidia/Morocco; 03–06 March 2016Google Scholar
- Baskonus HM, Bulut H (2015) On the numerical solutions of some fractional ordinary differential equations by fractional Adams–Bashforth–Moulton method. Open Math 13(1):547–556View ArticleGoogle Scholar
- Breaz D, Breaz N (2002) Two integral operators. Stud Univ Babes-Bolyai Math 47(3):13–19Google Scholar
- Breaz D, Owa S, Breaz N (2008) A new integral univalent operator. Acta Univ Apulensis Math Inform 16:11–16Google Scholar
- Breaz D, Güney HO, Salagean GS (2009) A new general integral operator. Tamsui Oxf J Math Sci 25(4):407–414Google Scholar
- Breaz D, Owa S, Breaz N (2014) Some properties for general integral operators. Adv Math 3(1):9–14Google Scholar
- Bulut H (2016) Analytical and numerical methods for solving nonlinear partial differential equations—I. In: 1st international symposium on computational mathematics and engineering sciences, Errichidia/Morocco, 03–06 March 2016Google Scholar
- Deniz E, Răducanu D, Orhan H (2012) On the univalence of an integral operator defined by Hadamard product. Appl Math Lett 25(2):179–184View ArticleGoogle Scholar
- Dixit KK, Chandra V (2008) On subclass of univalent functions with positive coefficients. Aligarh Bull Math 27(2):87–93Google Scholar
- El-Ashwah RM (2014) Subordination results for certain subclass of analytic functions defined by Salagean operator. Acta Univ Apulensis 37:197–204Google Scholar
- Esa Z, Kilicman A, Ibrahim RW, Ismail MR, Husain SS (2016a) Application of modified complex tremblay operator. In: Proceeding of 2nd international conference on mathematical sciences and statistics (ICMSS2016). 26–28 January, Kuala Lumpur, Malaysia, pp 26–28Google Scholar
- Esa Z, Srivastava HM, Kilicman A, Ibrahim RW (2016b) A novel subclass of analytic functions specified by a family of fractional derivatives in the complex domain. Filomat. http://arxiv.org/pdf/1511.01581.pdf
- Frasin BA (2011) Univalence criteria for general integral operator. Math Commun 16(1):115–124Google Scholar
- Kumar S, Kumar A, Baleanu D (2016) Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves. Nonlinear Dyn 1–17Google Scholar
- Miller SS, Mocanu P, Reade MO (1978) Starlike integral operators. Pac J Math 79(1):157–168View ArticleGoogle Scholar
- Najafzadeh S (2010) Application of Salagean and Ruscheweyh operators on univalent holomorphic functions with finitely many coefficients. Fract Calc Appl Anal 13(5):517–520Google Scholar
- Owa S, Srivastava HM (2002) Some generalized convolution properties associated with certain subclasses of analytic functions. J Inequal Pure Appl Math 3(3):1–13Google Scholar
- Pascu NN, Pescar V (1990) On the integral operators of Kim–Merkes and Pfaltzgraff. Mathematica (Cluj) 32(55):185–192Google Scholar
- Porwal S (2011) Mapping properties of an integral operator. Acta Univ Apulensis 27:151–155Google Scholar
- Shams S, Kulkarni SR, Jahangiri JM (2004) Classes of uniformly starlike and convex functions. Int J Math Math Sci 2004(55):2959–2961View ArticleGoogle Scholar
- Stanciu L, Breaz D (2014) Some properties of a general integral operator. Bull Iran Math Soc 40(6):1433–1439Google Scholar
- Sălăgean GS (1981) Subclasses of univalent functions, complex analysis-fifth Romanian–Finnish seminar, part 1 (Bucharest, 1981). Lect Notes Math 1013:362–372Google Scholar
- Yao JJ, Kumar A, Kumar S (2015) A fractional model to describe the Brownian motion of particles and its analytical solution. Adv Mech Eng 7(12):1687814015618,874View ArticleGoogle Scholar
- Yin XB, Kumar S, Kumar D (2015) A modified homotopy analysis method for solution of fractional wave equations. Adv Mech Eng 7(12):1687814015620330View ArticleGoogle Scholar