Is a function that is Strictly montonely increasing and Uniformly Continuous, with positive first derivative; and thrice differentiable
F:[0,1] to [0,1].
F(0)=0, F(1)=1, F(0.5)=0.5
is F alwas (1)strictly quasi convex, strictly quasi-concave?
if F is in addition, admits a continuous first derivative, is strictly positive
is F always (2) pseudo-convex and pseudo concave.? Or strictly so.
In effect, are these properties pseudo quasi (strict or not strict) effectively redundant once the function has been specified to be strictly increasing, and bounded, and continuous.? A
re these functions redudant, so long as F is defined on real domain and is strictly monotonely increasing (i know it implies quasi convexity, quasi concavity, and injectivity). Are such functions also schur convex?
Think about "S-shaped" function, like (shifted and rescaled) arctan(x) ...
I know there functions that are strictly monotonic but not pseudo convex or pseudo linear because they can have zero derivative , and thus zero derivative when there is no minima (as strict monotonic functions do not generally have them) but pseudoconvex or at least pseudo-linear functions only have zero first derivative or gradient at minima or maxima i think
Don't like arctan? Then try to make a graph of f(x) = a*((x-1/2)3 + (x-1/2)) +1/2 with a=4/5. It satisfies all your requirements but is neither convex nor concave (on [0,1]). Its derivative is strictly positive on [0,1].
i meant is is pseudo convex or pseudo concave? rather than concave or convex . ANd if not is is striclty quasi convex/concave etc . So i am speaking about these weaker versions not convexity or concavity
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