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On generalized preinvex functions and monotonicities. (English) Zbl 1094.26008
The generalization of preinvexity considered in this paper consists in replacing $\eta (x^{1},x^{2})$ by $\alpha (x^{1},x^{2})\eta (x^{1},x^{2})$ in the definition of invexity [see {\it S. R. Mohan} and {\it S. K. Neogy}, J. Math. Anal. Appl. 189, No. 3, 901--908 (1995; Zbl 0831.90097)], $\alpha $ being a given bifunction. The functions satisfying the resulting condition are said to be $\alpha $-preinvex with respect to $\eta $. Obviously, a function is $\alpha $-preinvex with respect to $\eta $ if and only if it is preinvex with respect to $\alpha \eta $. All results in the paper follow from this simple observation.

26B25Convexity and generalizations (several real variables)
26A48Monotonic functions, generalizations (one real variable)
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