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+--- Thread: 2 fixpoint pairs [2015] (/showthread.php?tid=959)
2 fixpoint pairs [2015] - tommy1729 - 02/18/2015
A general question.
Let A(z) be a transcendental real-entire function without real fixpoints.
Let I(z) be the functional inverse of A(z).
I(z) has no singularities for Re(z) > 0.
I(z) = z has exactly 2 conjugate pair solutions.
The fixpoints of I(z) are all repelling and have real part > 0.
Also they have the same derivative. (up to conjugate)
Then by using the Kneser method we can find 2 real-analytic superfunctions based on one of the fixpoint pairs of I(z).
By using analytic one-periodic theta functions we can modify those 2 superfunctions such that they are bounded in the strip 0 < Re(z) < 1.
Call these real-analytic superfunctions F(z) and G(z).
Consider D = lim F(z-n)/G(z-n)
Im intrested in the behaviour of
F(z) / ( D G(z) ) - 1
in the complex plane.
regards
tommy1729