Dear MHF members,
I have the following problem.
Problem. Let for ,
where with for all .
Prove or disprove that in (inside of ) implies in .
Thanks for your interest.
bkarpuz
Dear MHF members,
I have the following problem.
Problem. Let for ,
where with for all .
Prove or disprove that in (inside of ) implies in .
Thanks for your interest.
bkarpuz
Hi, I just gonna give some thoughts.
The region is a wedge below in the first quadrant.
The function integrates the function along the line .
It seems to me that the integrand could take negative values as long as it overshoots fast enough close to the region since you always have to integrate that part.
I think you can "build" such a function slice by slice ( by ). You could take some function that starts at zero, assume negative values until where it goes back to zero. Then it assumes large positive values from to . Find some way to make that -dependent, and you should have a counter-example. You could take for example a which period (and amplitude) varies with for the lower part and a sharp for the upper part.
I would think the result is true if . With the above construction, you see that we will always get into trouble close to the origin, i.e. we cannot find a function that will dive and overshoot fast enough for all since the domain of the line integral becomes infinitely small.
I think the construction still holds if you impose .
It seems the -dependence is crucial here. In fact, if you are given the -dependence of , you can make everything positive around the origin, and then do the dive-and-overshoot business once you have enough space (at larger ).