TPID 16
Let \( f(z) \) be a nonpolynomial real entire function.
\( f(z) \) has a conjugate primary fixpoint pair : \( L + M i , L - M i. \)
\( f(z) \) has no other primary fixpoints then the conjugate primary fixpoint pair.
For \( t \) between \( 0 \) and \( 1 \) and \( z \) such that \( Re(z) > 1 + L^2 \) we have that
\( f^{[t]}(z) \) is analytic in \( z \).
\( f^{[t]}(x) \) is analytic for all real \( x > 0 \) and all real \( t \ge 0 \) .
If \( f^{[t]}(x) \) is analytic for \( x = 0 \) then :
\( \frac{d^n}{dx^n} f^{[t]}(x) \ge 0 \) for all real \( x \ge 0 \) , all real \( t \ge 0 \) and all integer \( n > 0 \).
Otherwise
\( \frac{d^n}{dx^n} f^{[t]}(x) \ge 0 \) for all real \( x > 0 \) , all real \( t \ge 0 \) and all integer \( n > 0 \).
Are there solutions for \( f(z) \) ?
I conjecture yes.
regards
tommy1729
Let \( f(z) \) be a nonpolynomial real entire function.
\( f(z) \) has a conjugate primary fixpoint pair : \( L + M i , L - M i. \)
\( f(z) \) has no other primary fixpoints then the conjugate primary fixpoint pair.
For \( t \) between \( 0 \) and \( 1 \) and \( z \) such that \( Re(z) > 1 + L^2 \) we have that
\( f^{[t]}(z) \) is analytic in \( z \).
\( f^{[t]}(x) \) is analytic for all real \( x > 0 \) and all real \( t \ge 0 \) .
If \( f^{[t]}(x) \) is analytic for \( x = 0 \) then :
\( \frac{d^n}{dx^n} f^{[t]}(x) \ge 0 \) for all real \( x \ge 0 \) , all real \( t \ge 0 \) and all integer \( n > 0 \).
Otherwise
\( \frac{d^n}{dx^n} f^{[t]}(x) \ge 0 \) for all real \( x > 0 \) , all real \( t \ge 0 \) and all integer \( n > 0 \).
Are there solutions for \( f(z) \) ?
I conjecture yes.
regards
tommy1729