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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!

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