Prove the Theorem A below:

Let t be entire 1-periodic function, different from a constant, and
J(z)=z+t(z) for all complex z, and
let S be set of complex numbers with real part within segment [0,1].
Then: For any complex number p, except, perhaps, one value, in the domain S, there exist solution u of equation J(u)=p,
and this solution is not unique.

Prove the Theorem B below:

Let C be set of complex numbers z such that either |Im(z)|>0 or z>-2.
Then: There exist function F holomorphic on C such that
F(z^*)=F(z)^* for all z from C and
exp(F(z))=F(z+1) for all z from C
and F(0)=1.
There exist only one such function.

Make the Construction A below:

Let C be set of complex numbers such that Im(z)>0 or z>1.
Construct some function h holomorphic on C such that
h(z^*)=h(z)^* for all z from C and
h(h(z))=z! for all z from C, and
The growth of h(z) at large |z| is slower than exponential.
Suggest a way for evaluation of such a function.

Make the Construction B below:

For a given d>0, let B be set of complex numbers z such that at least one of the two conditions holds:
|z| < 1 or { Re(z)>0 , |Im(z)|<d } .
Let C be set of all complex numbers that do not belong to B.
Construct some entire function F, such that F(0)=1 and |F(z)|<|z| for all z in C.
Suggest a way for evaluation of such a function.