This will be a little experiment, in which the collaborative mathematics advocated by Timothy Gowers and others combines with my own frustration and laziness. If it goes well, I might try it more in the future.
Let p be a complex polynomial of degree d. Suppose that |p(z)|≤1 for all z such that |z|=1 and |z-1|≥δ (for some small δ>0). Then what’s the best upper bound you can prove on |p(1)|?
Note: I can prove an upper bound of the form |p(1)|≤exp(δd)—indeed, that holds even if p can be a polynomial in both z and its complex conjugate (and is tight in that case). What really interests me is whether a bound of the form |p(1)|≤exp(δ2d) is true.
Update: After I accepted Scott Morrison’s suggestion to post my problem at mathoverflow.net, the problem was solved 11 minutes later by David Speyer, using a very nice reduction to the case I’d already solved. Maybe I should feel sheepish, but I don’t—I feel grateful. I am now officially a fan of mathoverflow. Go there and participate!