  In a recent article (2022) we proved with L. Zaj\'i\v{c}ek that if $
G\subset\R^n $ is an unbounded open convex set that does not contain a
translation of a convex cone with non-empty interior, then there exist $
f:G\to\R $ and a concave modulus $ \omega $ such that $
\lim_{t\to\infty}\omega(t)=\infty $, $ f $ is both semiconvex and semiconcave
with modulus $ \omega $ and $ f\notin C^{1,\omega}(G) $. Here we improve the
previous result as follows: If $ G $ is as above and $ \omega(t)=t^{\alpha} $
for some $ \alpha\in(0,1) $, then there exists $ f:G\to\R $ that is both
semiconvex and semiconcave with modulus $ \omega $ and $ f\notin
C^{1,\alpha}(G) $. This result has immediate consequences concerning a
first-order quantitative converse Taylor theorem and the problem whether $ f\in
C^{1,\alpha}(G) $ whenever $ f $ is smooth in a corresponding sense on all
lines.
