A class of superordination-preserving convex integral operator

Saibah Siregar, Maslina Darus, Teodor Bulboacǎ

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

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/zγ0 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.

Original languageEnglish
Pages (from-to)379-390
Number of pages12
JournalMathematical Communications
Volume14
Issue number2
Publication statusPublished - Dec 2009

Fingerprint

Integral Operator
Small Function
Differential Subordination
Space of Analytic Functions
Subordination
Unit Disk
Denote
Imply
Generalise
Sufficient Conditions
Class

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

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

In: Mathematical Communications, Vol. 14, No. 2, 12.2009, p. 379-390.

Research output: Contribution to journalArticle

Siregar, S, Darus, M & Bulboacǎ, T 2009, 'A class of superordination-preserving convex integral operator', Mathematical Communications, vol. 14, no. 2, pp. 379-390.
Siregar, Saibah ; Darus, Maslina ; Bulboacǎ, Teodor. / A class of superordination-preserving convex integral operator. In: Mathematical Communications. 2009 ; Vol. 14, No. 2. pp. 379-390.
@article{26c9e4d331af40c89ec6ac52c3ef3074,
title = "A class of superordination-preserving convex integral operator",
abstract = "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.",
keywords = "Analytic function, Differential operator, Differential subordination, Starlike and convex function",
author = "Saibah Siregar and Maslina Darus and Teodor Bulboacǎ",
year = "2009",
month = "12",
language = "English",
volume = "14",
pages = "379--390",
journal = "Mathematical Communications",
issn = "1331-0623",
publisher = "Udruga Matematicara Osijek",
number = "2",

}

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 -