Fan Zheng - what theorem do you mean exactly?

They give the reference to the original book of Carlson with all proofs, please note it:B. C. Carlson (1977b). Special Functions of Applied Mathematics. New York: Academic Press.

the problem estimate is true for real $0<k<1$ and for imaginary $ik$ with $0<k<1$. If it will be proved also for $|k|=1$ it will be true for all $k$ due to the max principle for the corner, as it will be true on its three parts of boundary.

can we prove it for $|k|=1$?

May be here:proxy.math.rsu.ru/cgi-bin/library/…

Note that many references in wiki is not active/ Also there is not a reference to the best book on the subject in Russian: Komarov,Ponamarev,Slavyanov. Spheroidal and Coulomb Spheroidal Functions, 1976.

In the book of Gelfond there is a special chapter on convergence of Newton series.

The AGM inequality gives some trivial lower bound, not so?

The AGM inequality gives an upper bound with sum instead of the product. Are there a close form for this sum? By all means many terms in this sum are cancelled, is not it possible to find the close form for the rest?

may be a trivial AGM inequality is enough?

May be add "x" and change $\sin(n!)$ to $\sin(n!x)$ ? In this form it is a so called lacunar Fourier series, there are a lot of results on convergence for them. May be some result will help?

You mean a book "Analytic Number Theory"?

So the paper of J.H.E. Cohn contains solution to the Catalan Conjecture?