Let f be the fake half iterate of exp^[0.5].
About post nr 48 , I was thinking about stable polynomials and stable Taylor series.
Most theorems about roots however are about simple polynomials or relate a function with its derivative.
Notice that a function like 1 + x + x^2 + x^3 does have roots with a nonnegative real part.
Also 2 x^3 + 5 x^2 + 7 x + 31 = 0 has roots with strict positive real parts. Hence the situation is not trivial.
Certainly info about a functions derivative affects the position of its zero's , but then that info is often about the zero's of the derivative and therefore almost circular reasoning.
Now if one could say there is Always a positive integer M such that :
| a_M x^M | > f(x)/2.
where a_M is the M th Taylor coëfficiënt ,
that would be helpfull. But probably that is not the case here ??
At least not when x has the absolute value of one of the zero's on the negative line !!
Another way might be to show | f(x + yi) | > |f(x)| for x > 0.
But that is similar to a previous post.
I even considered the " pseudoinvariants " : f(ln(f)+1) , f(ln(f)+i) but with no success sofar.
Im running out of ideas.
These tetration type functions seem immume to many polynomial ideas despite the hadamard product form being one.
Its not immediately Obvious if theorems about polynomials can be extended to entire functions Q_n(z) that satisfy :
|Q_n(z)| < | C exp(z) |.
I Always liked Jensen's theorem about the roots of a derivative.
regards
tommy1729
About post nr 48 , I was thinking about stable polynomials and stable Taylor series.
Most theorems about roots however are about simple polynomials or relate a function with its derivative.
Notice that a function like 1 + x + x^2 + x^3 does have roots with a nonnegative real part.
Also 2 x^3 + 5 x^2 + 7 x + 31 = 0 has roots with strict positive real parts. Hence the situation is not trivial.
Certainly info about a functions derivative affects the position of its zero's , but then that info is often about the zero's of the derivative and therefore almost circular reasoning.
Now if one could say there is Always a positive integer M such that :
| a_M x^M | > f(x)/2.
where a_M is the M th Taylor coëfficiënt ,
that would be helpfull. But probably that is not the case here ??
At least not when x has the absolute value of one of the zero's on the negative line !!
Another way might be to show | f(x + yi) | > |f(x)| for x > 0.
But that is similar to a previous post.
I even considered the " pseudoinvariants " : f(ln(f)+1) , f(ln(f)+i) but with no success sofar.
Im running out of ideas.
These tetration type functions seem immume to many polynomial ideas despite the hadamard product form being one.
Its not immediately Obvious if theorems about polynomials can be extended to entire functions Q_n(z) that satisfy :
|Q_n(z)| < | C exp(z) |.
I Always liked Jensen's theorem about the roots of a derivative.
regards
tommy1729
