数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (3): 989-1001.doi: 10.1016/S0252-9602(12)60074-5

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SANDWICH-TYPE RESULTS FOR A CLASS OF CONVEX INTEGRAL OPERATORS

Teodor Bulboacä   

  1. Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, 400084 Cluj-Napoca, Romania
  • 收稿日期:2010-05-04 修回日期:2011-02-14 出版日期:2012-05-20 发布日期:2012-05-20

SANDWICH-TYPE RESULTS FOR A CLASS OF CONVEX INTEGRAL OPERATORS

Teodor Bulboacä   

  1. Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, 400084 Cluj-Napoca, Romania
  • Received:2010-05-04 Revised:2011-02-14 Online:2012-05-20 Published:2012-05-20

摘要:

Let H(U) be the space of analytic functions in the unit disk U. For the integral operator AΦ,φα,β,ν : KH(U), with KH(U), defined by
AΦ,φα,β,ν[f](z) =[β+ν/zνΦ(z)∫z0f (t)φ(t)tδ−1dt]1/ β,
where α, βνδ ∈C and ΦφH(U), we will determine sufficient conditions on g1, g2α, β and ν, such that
(z)[g1(z)/z]α< (z)[ f(z)/z]α< zφ(z)[g2(z)/z]α
implies
(z)[AΦ,φα,β,ν[g1](z)/z]β< (z)[AΦ,φα,β,ν[f](z)/z]β< (z)[AΦ,φα,β,ν[g2](z)/z]β
.
The symbol “<” stands for subordination, and we call such a kind of result a sandwich-type theorem. In addition, (z)[AΦ,φα,β,ν[[g1](z)/z]β is the largest function and (z)[AΦ,φα,β,ν[g2](z)/z]β the smallest function so that the left-hand side, respectively the right-hand side of the above implications hold, for all f functions satisfying the assumption. We give a particular case of the main result obtained for appropriate choices of functions Φ and φ that also generalizes classic results of the theory of differential subordination and superordination.

关键词: Analytic function, starlike and convex function, differential operator, differ-ential subordination

Abstract:

Let H(U) be the space of analytic functions in the unit disk U. For the integral operator AΦ,φα,β,ν : KH(U), with KH(U), defined by
AΦ,φα,β,ν[f](z) =[β+ν/zνΦ(z)∫z0f (t)φ(t)tδ−1dt]1/ β,
where α, βνδ ∈C and ΦφH(U), we will determine sufficient conditions on g1, g2α, β and ν, such that
(z)[g1(z)/z]α< (z)[ f(z)/z]α< zφ(z)[g2(z)/z]α
implies
(z)[AΦ,φα,β,ν[g1](z)/z]β< (z)[AΦ,φα,β,ν[f](z)/z]β< (z)[AΦ,φα,β,ν[g2](z)/z]β
.
The symbol “<” stands for subordination, and we call such a kind of result a sandwich-type theorem. In addition, (z)[AΦ,φα,β,ν[[g1](z)/z]β is the largest function and (z)[AΦ,φα,β,ν[g2](z)/z]β the smallest function so that the left-hand side, respectively the right-hand side of the above implications hold, for all f functions satisfying the assumption. We give a particular case of the main result obtained for appropriate choices of functions Φ and φ that also generalizes classic results of the theory of differential subordination and superordination.

Key words: Analytic function, starlike and convex function, differential operator, differ-ential subordination

中图分类号: 

  • 30C80