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### Reference(s) on the smallest concave majorant for the sequence of coefficients of a given power series?

A better reference on Valiron is George Valiron, Fonctions analytiques, Presses universitaires de France, 1954, MR0061658, Zbl 0055.06702. This smallest concave majorant is nothing but the Newton ...
You can use the pre-$\lambda$ ring identity $$\big(\sum_n [\mathrm{Sym}^n(X)]t^n\big)\big( \sum_k \lambda^k[X](-t)^k\big) = 1.$$ So it is enough to show that the generating series for the exterior ...
### Can one find an elementary function $f(t)$ such that ${}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)=f(t)$?
Maple does it in terms of complete elliptic integrals $\rm{K}$ and $\rm{E}$ ...  {\mbox{$_2$F$_1$}\left(\frac12,\frac12;\,2;\,t\right)}={\frac {4\left( t-1 \right){\rm K} \left( \sqrt {t} \right) ...