See attached document page 14;
the equation expressed by sublinear functions when convex. Do they mean midpoint convexity here?. Likwise am i correct taht 2F(x)=f(2x) is not usually a property of midpoint convexity or full convexity, even if F(0)=0, F(1)=1, F(1/2) unless other properties such as symmetry or sub-linearity are imposed (which I have noticed)
Am i correct in presuming that by convex that mean just mid point convex. At the same time sub-linear functions appear to be asserting something very much closer to a homogeneity condition that cauchy equations do not have without continuity. Ie it appears peculiar that a sublinear function appear to satisfy (f(sigmax)=sigma(F(x)) for at least the positive rationals whilst only satisfying sub-additivity? I can only presume that such functions either do satisfy full functional additivity for rational numbers, but are not homogeneous with regard to irrationals when made continuous, or that they satisfy essentially nothing at until, they are continuous, whereapon such point they arguably very similar to linear functions (and where they essentially possess full additivity over (positive )rationals (or only integers?), but only when continuous, whilst their homogeineity properties are very similar to linear functions, and otherwise only differ in that they are subadditive with regard to irrationals (and some rationals, presumably negative values, which may be often irrelevant), and presumable neither additive or sub-additive with regard to rational or irrationals, unless continuous (unlike cauchys equation F(x+y)=F(x)+F(y)
. Although what is expressed there is arguably stronger, it appears to have a property of jensens equality (F(x/2+y/2)=F(x)/2+F(y)/2 , ie F(2x)=2F(x) for F(0)=0, whilst not completely satisfying it at the same time (it only satisfy the midpoint convexity; the first half of jensens equation at the same time); ie satisfies midpoint convexity;
F(x/2+y/2)
I'm sorry. This question is not in sphere of my interesting.
Good luck and my best wishes,
Regards, Mirjana
IT appears that if a model is already rather symmetric; then midpoint convexity appears to entail midpoint concavity; and thus jensen's equation; at least pair for pair for all points, and has the same functional results.
I presume then that most symmetric functions such that
1.at least functions F :F: [0,1] to [0,1] closed and bounded
2.twice differentiable; F''(0)=0 if need be; F'(x) positive
3 Convex; and at least (both 'quasi concave and strictly quasi concave'); a
4. continuous given convexity; and thus lipschitz continuous given convexity on the closed bounded unit interval domain [0,1] and co-domain apparently which are also
5..strictly monotone increasing;
6. F(0.5)=0.5; F(0)=0 and F(1)=1; all x in (0,1) 01;x1+x21;y1+y2
Dear Professor Ger, I think that this is a very good good suggestion !. As my posts become quite unreadable and long winded otherwise. I am particularly interested in relationship between said models on the unit interval, and when convex and continuous (if they are convex with F(0)=0, F(1)=1; and strictly monotonic, wont said models be super-additive as well?. If positive homogeineity pertain to a all reals numbers, cant one get to cauchy equations simply from that-- these are similar to von neumanns symmetric gauge models which are necessarily convex continuous F(0)=0, positively homogeneious, positive and sub-additive as well). The issue I am having is that I have noticed that my model (is symmetric and given midpoint convexity it seems to entail midpoint concavity and thus jensens equation). I will write this out more formally and email you. I am very grateful for this!
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