### Abstract

If H(U) denotes the space of analytic functions in the unit disk U, for the integral operator A_{α,β,γ}
^{h} :K→ -H(U), with K⊂H(U), defined by A _{α,β,γ,δ}
^{h}[f](z) = [β+γ/z^{γ} ∫_{0}
^{x} f ^{α}(t)h(t)t^{δ-1}dt]^{1/β}, (α, β, γ, δ, ∈, C and h) ∈ H(U), we will determine sufficient conditions on, g_{2}, α, β and γ such that zh(z) [g_{1}(z)/z]^{α} ≺ zh(z)[f(z)/z] ^{α} ≺[g_{2}(z)/z]^{α} implies z[A _{α,β,γ,δ}
^{h}[g_{1}(z)/z] ^{β} z [ A_{α,β,γ,δ} ^{h}[g](z)/z]β [A_{α,β,γ,δ} ^{h}[g_{2}(z)/z]β. In addition, both of the subordinations are sharp, since the left-hand side will be the largest function, and the right-hand side will be the smallest function so that the above implication has been held for all f functions satisfying the double differential subordination of the assumption. The results generalize those of the last author from [3], obtained for the special case α = β and h ≡ l.

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

Pages (from-to) | 379-390 |

Number of pages | 12 |

Journal | Mathematical Communications |

Volume | 14 |

Issue number | 2 |

Publication status | Published - Dec 2009 |

### Fingerprint

### Keywords

- Analytic function
- Differential operator
- Differential subordination
- Starlike and convex function

### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Applied Mathematics
- Geometry and Topology

### Cite this

*Mathematical Communications*,

*14*(2), 379-390.

**A class of superordination-preserving convex integral operator.** / Siregar, Saibah; Darus, Maslina; Bulboacǎ, Teodor.

Research output: Contribution to journal › Article

*Mathematical Communications*, vol. 14, no. 2, pp. 379-390.

}

TY - JOUR

T1 - A class of superordination-preserving convex integral operator

AU - Siregar, Saibah

AU - Darus, Maslina

AU - Bulboacǎ, Teodor

PY - 2009/12

Y1 - 2009/12

N2 - If H(U) denotes the space of analytic functions in the unit disk U, for the integral operator Aα,β,γ h :K→ -H(U), with K⊂H(U), defined by A α,β,γ,δ h[f](z) = [β+γ/zγ ∫0 x f α(t)h(t)tδ-1dt]1/β, (α, β, γ, δ, ∈, C and h) ∈ H(U), we will determine sufficient conditions on, g2, α, β and γ such that zh(z) [g1(z)/z]α ≺ zh(z)[f(z)/z] α ≺[g2(z)/z]α implies z[A α,β,γ,δ h[g1(z)/z] β z [ Aα,β,γ,δ h[g](z)/z]β [Aα,β,γ,δ h[g2(z)/z]β. In addition, both of the subordinations are sharp, since the left-hand side will be the largest function, and the right-hand side will be the smallest function so that the above implication has been held for all f functions satisfying the double differential subordination of the assumption. The results generalize those of the last author from [3], obtained for the special case α = β and h ≡ l.

AB - If H(U) denotes the space of analytic functions in the unit disk U, for the integral operator Aα,β,γ h :K→ -H(U), with K⊂H(U), defined by A α,β,γ,δ h[f](z) = [β+γ/zγ ∫0 x f α(t)h(t)tδ-1dt]1/β, (α, β, γ, δ, ∈, C and h) ∈ H(U), we will determine sufficient conditions on, g2, α, β and γ such that zh(z) [g1(z)/z]α ≺ zh(z)[f(z)/z] α ≺[g2(z)/z]α implies z[A α,β,γ,δ h[g1(z)/z] β z [ Aα,β,γ,δ h[g](z)/z]β [Aα,β,γ,δ h[g2(z)/z]β. In addition, both of the subordinations are sharp, since the left-hand side will be the largest function, and the right-hand side will be the smallest function so that the above implication has been held for all f functions satisfying the double differential subordination of the assumption. The results generalize those of the last author from [3], obtained for the special case α = β and h ≡ l.

KW - Analytic function

KW - Differential operator

KW - Differential subordination

KW - Starlike and convex function

UR - http://www.scopus.com/inward/record.url?scp=74549161562&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=74549161562&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:74549161562

VL - 14

SP - 379

EP - 390

JO - Mathematical Communications

JF - Mathematical Communications

SN - 1331-0623

IS - 2

ER -