Conjecture

The following holds for regular tetration:

for all and 0 < |y| < 1.

The following holds for intuitive tetration:

for all and 0 < |y| < 1.

Discussion

This would be interesting in its own right, partly because it is symmetric, but also because it is useful in demonstrating the difference between reciprocal heights and super-roots. Another notable aspect of this set of bounds is that it is very extension-dependent. This does not hold for linear tetration.

Appended are several graphs of the function for x = 1.001, 1.1, eta, e, 10. The first three (1.001, 1.1, eta) were calculated with regular tetration, and the last two (e, 10) were calculated with intuitive tetration. More discussion of this is here

The following holds for regular tetration:

for all and 0 < |y| < 1.

The following holds for intuitive tetration:

for all and 0 < |y| < 1.

Discussion

This would be interesting in its own right, partly because it is symmetric, but also because it is useful in demonstrating the difference between reciprocal heights and super-roots. Another notable aspect of this set of bounds is that it is very extension-dependent. This does not hold for linear tetration.

Appended are several graphs of the function for x = 1.001, 1.1, eta, e, 10. The first three (1.001, 1.1, eta) were calculated with regular tetration, and the last two (e, 10) were calculated with intuitive tetration. More discussion of this is here