### Abstract

Let q_{1}, q_{2} be univalent in Δ:= {z: |z| < 1} and p be certain analytic function. We give some applications of first order differential subordinations and superordinations to obtain sufficient conditions to satisfy the following sandwich implication which is a generalization for various known sandwich theorems: βzq_{1}
^{k}(z) q̀_{1}(z) + ∑_{j=0}
^{n} α_{j} q_{1}
^{j}(z) ≺ βzp^{k}(z)p̀(z) + ∑_{j=0}
^{n} α_{j} p^{j}(z) ≺ βzq_{2}
^{k}(z)q̀_{2}(z) + ∑_{j=0}
^{n} α_{j} q_{2} ^{j}(z)implies q_{1}(z) ≺ p(z) ≺ q_{2}(z), where k∈ ℤ and β ≠ 0, ὰ_{j} s ∈ ℂ. Some of its special cases and its applications will be considered for certain analytic functions and certain linear operators.

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

Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Lobachevskii Journal of Mathematics |

Volume | 30 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2009 |

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### Keywords

- Differential subordinations
- Differential superordinations
- Dominant
- Subordinant

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Lobachevskii Journal of Mathematics*,

*30*(1), 1-11. https://doi.org/10.1134/S1995080209010016

**Main differential sandwich theorem with some applications.** / Al-Refai, Oqlah; Darus, Maslina.

Research output: Contribution to journal › Article

*Lobachevskii Journal of Mathematics*, vol. 30, no. 1, pp. 1-11. https://doi.org/10.1134/S1995080209010016

}

TY - JOUR

T1 - Main differential sandwich theorem with some applications

AU - Al-Refai, Oqlah

AU - Darus, Maslina

PY - 2009/1

Y1 - 2009/1

N2 - Let q1, q2 be univalent in Δ:= {z: |z| < 1} and p be certain analytic function. We give some applications of first order differential subordinations and superordinations to obtain sufficient conditions to satisfy the following sandwich implication which is a generalization for various known sandwich theorems: βzq1 k(z) q̀1(z) + ∑j=0 n αj q1 j(z) ≺ βzpk(z)p̀(z) + ∑j=0 n αj pj(z) ≺ βzq2 k(z)q̀2(z) + ∑j=0 n αj q2 j(z)implies q1(z) ≺ p(z) ≺ q2(z), where k∈ ℤ and β ≠ 0, ὰj s ∈ ℂ. Some of its special cases and its applications will be considered for certain analytic functions and certain linear operators.

AB - Let q1, q2 be univalent in Δ:= {z: |z| < 1} and p be certain analytic function. We give some applications of first order differential subordinations and superordinations to obtain sufficient conditions to satisfy the following sandwich implication which is a generalization for various known sandwich theorems: βzq1 k(z) q̀1(z) + ∑j=0 n αj q1 j(z) ≺ βzpk(z)p̀(z) + ∑j=0 n αj pj(z) ≺ βzq2 k(z)q̀2(z) + ∑j=0 n αj q2 j(z)implies q1(z) ≺ p(z) ≺ q2(z), where k∈ ℤ and β ≠ 0, ὰj s ∈ ℂ. Some of its special cases and its applications will be considered for certain analytic functions and certain linear operators.

KW - Differential subordinations

KW - Differential superordinations

KW - Dominant

KW - Subordinant

UR - http://www.scopus.com/inward/record.url?scp=63549123787&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=63549123787&partnerID=8YFLogxK

U2 - 10.1134/S1995080209010016

DO - 10.1134/S1995080209010016

M3 - Article

AN - SCOPUS:63549123787

VL - 30

SP - 1

EP - 11

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 1

ER -