### Abstract

Let ℋ denote the class of functions f=h+ ḡ that are harmonic univalent and sense-preserv- ing in the unit disk U=z:|z|<1, where h(z)=z+ ∑ k=2 ∞ ak zk, g(z)= ∑ k=1 ∞ bk zk ( | b1 |<1) . In this paper, we introduce the class Mℋ ( n,λ,α) of functions f=h+ ḡ which are harmonic in U. A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the class M ℋ̄ ( n,λ,α) if fn (z)=h+ gn ̄ ∈Mℋ ( n,λ,α), where h(z)=z- ∑ k=2 ∞ | ak | zk, gn (z)=( -1)n ∑ k=1 ∞ | bk |zk and n∈0 . Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the class M ℋ̄ ( n,λ,α), are obtained.

Original language | English |
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Article number | 263413 |

Journal | Journal of Inequalities and Applications |

Volume | 2008 |

DOIs | |

Publication status | Published - 2008 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*Journal of Inequalities and Applications*,

*2008*, [263413]. https://doi.org/10.1155/2008/263413

**On harmonic functions defined by derivative operator.** / Al-Shaqsi, K.; Darus, Maslina.

Research output: Contribution to journal › Article

*Journal of Inequalities and Applications*, vol. 2008, 263413. https://doi.org/10.1155/2008/263413

}

TY - JOUR

T1 - On harmonic functions defined by derivative operator

AU - Al-Shaqsi, K.

AU - Darus, Maslina

PY - 2008

Y1 - 2008

N2 - Let ℋ denote the class of functions f=h+ ḡ that are harmonic univalent and sense-preserv- ing in the unit disk U=z:|z|<1, where h(z)=z+ ∑ k=2 ∞ ak zk, g(z)= ∑ k=1 ∞ bk zk ( | b1 |<1) . In this paper, we introduce the class Mℋ ( n,λ,α) of functions f=h+ ḡ which are harmonic in U. A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the class M ℋ̄ ( n,λ,α) if fn (z)=h+ gn ̄ ∈Mℋ ( n,λ,α), where h(z)=z- ∑ k=2 ∞ | ak | zk, gn (z)=( -1)n ∑ k=1 ∞ | bk |zk and n∈0 . Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the class M ℋ̄ ( n,λ,α), are obtained.

AB - Let ℋ denote the class of functions f=h+ ḡ that are harmonic univalent and sense-preserv- ing in the unit disk U=z:|z|<1, where h(z)=z+ ∑ k=2 ∞ ak zk, g(z)= ∑ k=1 ∞ bk zk ( | b1 |<1) . In this paper, we introduce the class Mℋ ( n,λ,α) of functions f=h+ ḡ which are harmonic in U. A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the class M ℋ̄ ( n,λ,α) if fn (z)=h+ gn ̄ ∈Mℋ ( n,λ,α), where h(z)=z- ∑ k=2 ∞ | ak | zk, gn (z)=( -1)n ∑ k=1 ∞ | bk |zk and n∈0 . Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the class M ℋ̄ ( n,λ,α), are obtained.

UR - http://www.scopus.com/inward/record.url?scp=41149173189&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41149173189&partnerID=8YFLogxK

U2 - 10.1155/2008/263413

DO - 10.1155/2008/263413

M3 - Article

AN - SCOPUS:41149173189

VL - 2008

JO - Journal of Inequalities and Applications

JF - Journal of Inequalities and Applications

SN - 1025-5834

M1 - 263413

ER -