### Abstract

By using the fractional derivative operator of Owa and Srivastava, we define a new linear multiplier fractional differential operator. Some generalized classes of analytic functions containing this multiplier are introduced. Basic properties of these classes are studied, such as inclusion relations and coefficient bounds. Some well known subclasses are pointed out as new special cases of our results. Moreover, the Cesáro partial sums σ _{m} of functions f are considered, and sharp lower bounds for the ratios of real part of f and σ _{m} (and also of f' and σ' _{m}) are determined in the unit open disk.

Original language | English |
---|---|

Pages (from-to) | 167-184 |

Number of pages | 18 |

Journal | Miskolc Mathematical Notes |

Volume | 12 |

Issue number | 2 |

Publication status | Published - 2011 |

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### Keywords

- Cesáro partial sums
- Close to convex function
- Convex function
- Fractional derivative
- Inclusion relations
- Starlike function
- Subordination
- Superordination

### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Control and Optimization
- Discrete Mathematics and Combinatorics
- Numerical Analysis

### Cite this

*Miskolc Mathematical Notes*,

*12*(2), 167-184.

**Differential operator generalized by fractional derivatives.** / Ibrahim, Rabha W.; Darus, Maslina.

Research output: Contribution to journal › Article

*Miskolc Mathematical Notes*, vol. 12, no. 2, pp. 167-184.

}

TY - JOUR

T1 - Differential operator generalized by fractional derivatives

AU - Ibrahim, Rabha W.

AU - Darus, Maslina

PY - 2011

Y1 - 2011

N2 - By using the fractional derivative operator of Owa and Srivastava, we define a new linear multiplier fractional differential operator. Some generalized classes of analytic functions containing this multiplier are introduced. Basic properties of these classes are studied, such as inclusion relations and coefficient bounds. Some well known subclasses are pointed out as new special cases of our results. Moreover, the Cesáro partial sums σ m of functions f are considered, and sharp lower bounds for the ratios of real part of f and σ m (and also of f' and σ' m) are determined in the unit open disk.

AB - By using the fractional derivative operator of Owa and Srivastava, we define a new linear multiplier fractional differential operator. Some generalized classes of analytic functions containing this multiplier are introduced. Basic properties of these classes are studied, such as inclusion relations and coefficient bounds. Some well known subclasses are pointed out as new special cases of our results. Moreover, the Cesáro partial sums σ m of functions f are considered, and sharp lower bounds for the ratios of real part of f and σ m (and also of f' and σ' m) are determined in the unit open disk.

KW - Cesáro partial sums

KW - Close to convex function

KW - Convex function

KW - Fractional derivative

KW - Inclusion relations

KW - Starlike function

KW - Subordination

KW - Superordination

UR - http://www.scopus.com/inward/record.url?scp=84857412082&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857412082&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84857412082

VL - 12

SP - 167

EP - 184

JO - Miskolc Mathematical Notes

JF - Miskolc Mathematical Notes

SN - 1787-2405

IS - 2

ER -