Real-analytic tetration uniqueness criterion?

[sheldonison]

Yes, I guess that's another conjecture, that for all real(z)>~=0.5, if you make a line from -imag infinity to +imag(infinity) at real(z), the maximum absolute value occurs at the real axis. This is also supported by empirical evidence, but I can't think of any obvious way to prove it. This would also mean that the maximum magnitude on any circle on the real axis occurs at the real axis, so long as its bigger than about 0.5.

I did some experiments and this holds if z>=z0 where z0=0.47823520737667784466. What is special about this particular number, z0? Well, sexp''(z0-1)=0, which means that \( \Re(\text{sexp}(z0-1+k i))\approx 0.4777430947666662352 -0.021068926393682 k^4 \) near the real axis, where these are the a0 and a4 Taylor series coefficients.