# 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

## 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.

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## Authors’ Affiliations

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