Tetration Forum

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I would like to say im not a huge fan of the usual definition used sexp(-2) = -oo.

So i use two new functions resp definitions that are just a " shift ".

Exp^[a](-00) = newsexp(a)

Now newsexp(0) = -00 and
Newslog(-00) = 0.

Exp^[a](x) is still newsexp(newslog(x) + a).

And newsexp is similar to ln , newslog is similar to exp.

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As Said in the title a ( returning ?) question that should perhaps be posted as TPID 19 in the open problems section.

Also clearly related to the above.

Consider all C^oo solutions to f(x) = exp[1/2](x).

Now consider the subset of those that satisfy

For all real x : f ' (x) , f " (x) > 0.

Then what are the max and min values of

f ( - oo ).

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Although approximations exist , i am unaware of a good method , both in theory and numerical / practical.

No closed form known to me , not even with tet type functions.

If someone conjectured a closed form , i have no good method to consider proof or disproof ( only luck from iterations ).

It is pointless to add links to related subjects here , because almost all of them are !

So , therefore , I consider it a key question , perhaps worthy of a TPID 19.

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Regards

Tommy1729