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