Is there a name for these properties. A function f(x, y) defined for x > 0 and y > 0 is (strictly or weakly) increasing on all rays through the origin; i.e, for all x, y, k > 0, f(kx, ky) > (or >=) f(x, y). If it helps you can assume the function is continuous or differentiable.
Take a look at monotonic functions
Thanks for making me feel real stupid. Stupid though I may be I don’t like it pointed out. 
(Saw the above as the “Moronic” Functions…which of course reflects on me and not the math.)
Monotonicity would require that f(x, y) < f(x’, y’) whenever x < x’ and y < y’. This is, or at least appears to be, a strictly weaker property.
It’s equivalent. If you increase whenever x increases, and you increase whenever y increases, then you increase whenever x and y both increase, and vice versa (this is true both for the weak interpretation of increasing, and for the strict interpretation of increasing (if “x and y both increase” is taken to mean “x and y both weakly increase and at least one strictly increases”)).
If k can be any value > 0, then all values on a ray must be the same. Not that it doesn’t make any interesting functions, but I think something like monotonicity may be what’s desired.
(Proof: Let f(x,y) = a.
For k = m, f(mx,my) = b, b >= a.
For k = 1/m, f(mx/m, my,y) >= b.
f(mx/m,my/m) = f(x,y) = a.
Thus a >= b, and b >= a, implying a = b.)
First off I must apologize for a misstatement in my original post. If the function is increasing on rays through the origin, then we need k > 1 not k > 0 in my further description. But …
No it is not the same as monotonicity they’re not equivalent. Monotonicity implies the property I want but not vice versa. For example, consider the function x[sup]2[/sup]/y. Clearly this is not a monotone function as it is decreasing in y, but f(kx, ky) = kf(x, y) > f(x,y) for k > 1 so it satisfies my property.
This looks a lot like some form of homogeneity.
It is a bit like it, but homogeneity is a special case. It’s close to an increasing homothetic function, but I’m not sure it’s that either. An increasing homothetic function is G(f(x,y)) where f(x,y) is homogeneous function of degree a >0 (or log homogeneous) and G( ) is a monotonic increasing function.
Oh, whoops, I was thinking all rays through any point, not all rays through the origin. (There was also some confusion about what you intended for rays with negative X or Y component.) :smack:
Well, it’s basically a function which is increasing with respect to r in polar coordinates. Is that snappy enough?
No negative components for x or y per my original statement. And I perfectly willing to assume f(x, y) > 0 if that helps.
For those still interested, the answer appears to be homothetic and increasing. I knew homotheticity and increasing implied this propoerty but by an alternate definition I’ve just found it appears they are the same.
The definition, I always knew of homotheticity is f(x,y) is homothetic if f(x,y) = G(g(x,y)) with g() homogeneous of any degree and G() monotonic. It seems an alternate definition is f(x,y) < f(x’,y’) iff f(kx,ky) < f(kx’,ky’) for all k > 0. That is the isoquants of the function are all radial expansions of each other.
Then we just need to add that the function increases as we go out from the origin.