09/02/2007, 10:24 PM
I think I've figured out where my maths went wrong, with respect to my attempt to require the second derivative of the logarithm to be positive. I'll attempt to redo the maths from the start with Henryk's suggested notation, to prevent any confusion and hopefully make it more rigorous. However, this may unfortunately have to wait a day or two, as doing that maths and writing the TeX code takes some time for me, and I'll be busy most of the rest of today, and at the beach for Labor Day tomorrow.
Assuming I can fix this up, the next challenge is then to see if I can perform the same "magic" with a derivative of E(x). So far I've tried to show that E(x) is convex. Now I want to see if I can show what is required for E'(x) and/or E''(x) to be convex. (I may skip to the second derivative, depending on what happens with negative iterates, which have alternating signs/curvatures for their derivatives.)
Assuming I can fix this up, the next challenge is then to see if I can perform the same "magic" with a derivative of E(x). So far I've tried to show that E(x) is convex. Now I want to see if I can show what is required for E'(x) and/or E''(x) to be convex. (I may skip to the second derivative, depending on what happens with negative iterates, which have alternating signs/curvatures for their derivatives.)
~ Jay Daniel Fox