A function f is quasiconvex if and only if f(δ x + (1 - δ y) ≤ max{f(x),f y)} for any δ ∈ [0,1], and the negative of a quasiconvex function is quasiconcave[20].
The lsc function (f:Xrightarrowmathbb{R} cup{+infty}) is said to be a quasiconvex function, if for any (x,y in X), one has f z leqmaxbigl{ f(x),f y bigr}, quadforall x,y in X, zin[x,y], where ([x,y]={x+t y-x) : tin(0,1)}).
The objective function κ is a quasiconvex function [ 22].
Lemma 21 Let C be a boundedly weakly compact and convex subset of a Banach space E. Let T be a mapping on a subset C. Assume that there exists a lower semicontinuous, strictly quasiconvex function f from C into ℝ such that f is nonincreasing with respect to T and f satisfies (4).
Lemma 23 Let C be a weakly compact and convex subset of a Banach space E. Let S = { T 0 } ∪ S ′ be a family of commuting mappings on C. Assume that for every T ∈ S, there exists a lower semicontinuous, strictly quasiconvex function f T from C into ℝ such that f T is nonincreasing with respect to T. Assume also that F ( T ) is closed and convex for T ∈ S ′.
Lemma 22 Let C be a boundedly weakly compact and convex subset of a Banach space E. Let T 0, T 1, T 2, …, T ℓ be commuting mappings on C. Assume that for every j = 0, 1, 2, …, ℓ, there exists a lower semicontinuous, strictly quasiconvex function f j from C into ℝ such that f j is nonincreasing with respect to T j and f j satisfies (4).
(i) (Dutta and Chandra [24]) If f is a differentiable function and ∂*f(x̅) is an upper regular convexifactor, then ∂*f(x̅) = {∇f(x̅)}, and the above definition reduces to the definition of quasiconvex function.
These subclasses extend the classical subclasses of meromorphic starlike, convex, close-to-convex, and quasiconvex functions.
Clearly the classes and extend the classical classes of close-to-convex and quasiconvex functions, respectively.
In fact, the real-valued arcwise connected quasiconvex functions had already appeared in early works, such as [1].
