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Full Version: 2 fixpoint pairs [2015]
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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.