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Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 12736, 13 pages

Generalized preinvex functions and their properties

Department of Mathematics, COMSATS Institute of Information Technology, Plot 30, Sector H-8, Islamabad, Pakistan

Received 9 January 2005; Revised 9 August 2005; Accepted 2 September 2005

Copyright © 2006 Muhammad Aslam Noor and Khalida Inayat Noor. 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.


We introduce some new classes of preinvex and invex functions, which are called ϕ-preinvex and ϕ-invex functions. We study some properties of these classes of ϕ-preinvex (ϕ-invex) functions. In particular, we establish the equivalence among the ϕ-preinvex functions, ϕ-invex functions, and ϕη-monotonicity of their differential under some suitable conditions.