International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 275935, 12 pages
doi:10.1155/2010/275935
Research Article

On Certain Classes of 𝑝 -Valent Functions by Using Complex-Order and Differential Subordination

Abdolreza Tehranchi1 and Adem Kılıçman2

1Department of Mathematics, Islamic Azad University, South Tehran Branch, Tehran, Iran
2Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM), Serdang, Selangor 43400, Malaysia

Received 29 May 2010; Revised 24 September 2010; Accepted 16 October 2010

Academic Editor: Vladimir Mityushev

Copyright © 2010 Abdolreza Tehranchi and Adem Kılıçman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of the present paper is to study the 𝑝 -valent analytic functions in the unit disk and satisfy the differential subordinations 𝑧 ( ℐ 𝑝 ( 𝑟 , 𝜆 ) 𝑓 ( 𝑧 ) ) ( 𝑗 + 1 ) / ( 𝑝 − 𝑗 ) ( ℐ 𝑝 ( 𝑟 , 𝜆 ) 𝑓 ( 𝑧 ) ) ( 𝑗 ) ≺ ( 𝑎 + ( 𝑎 𝐵 + ( 𝐴 − 𝐵 ) 𝛽 ) 𝑧 ) / 𝑎 ( 1 + 𝐵 𝑧 ) , where 𝐼 𝑝 ( 𝑟 , 𝜆 ) is an operator defined by S ̆ ‌ a l ̆ ‌ a gean and 𝛽 is a complex number. Further we define a new related integral operator and also study the Fekete-Szego problem by proving some interesting properties.