exp^[1/2](x) uniqueness from 2sinh ?

[tommy1729]

A possible uniqueness critertion for exp^[1/2](x) ?

For x > 1 and any integer n >= 0 :

1) e/n! > d^n exp^[1/2](x)/d^n x @ x = 1 > 0.

2) 2sinh^[1/2](x) + d 2sinh^[1/2](x)/dx - exp(-x) > exp^[1/2](x) > 2sinh^[1/2](x).
( 2sinh^[1/2](x) is computed with the koenigs function )

3) exp^[1/2](z) is holomorphic for Re(z) > 1/2.

If the uniqueness fails the question is if the conditions are too strong or too weak.

And if it can be improved.

regards

tommy1729

[tommy1729]

Hmm The conditions must fail because they imply that exp^[1/2](x) is entire which it is not.

Not sure how to bound the derivatives then ...

reduce condition 1) to d^n exp^[1/2](x)/d^n x @ x = 1 > 0 ?

regards

tommy1729