  Our paper is a complement to a recent article by D. Azagra and C. Mudarra
(2021). We show how older results on semiconvex functions with modulus $\omega$
easily imply extension theorems for $C^{1,\omega}$-smooth functions on
super-reflexive Banach spaces which are versions of some theorems of Azagra and
Mudarra. We present also some new interesting consequences which are not
mentioned in their article, in particular extensions of $C^{1,\omega}$-smooth
functions from open quasiconvex sets. They proved also an extension theorem for
$C_{\mathrm B}^{1,+}$-smooth functions (i.e., functions with uniformly
continuous derivative on each bounded set) on Hilbert spaces. Our version of
this theorem and new extension results for $C^{1,+}$ and
$C_{\mathrm{loc}}^{1,+}$-smooth functions (i.e., functions with uniformly,
resp. locally uniformly continuous derivative), all of which are proved on
arbitrary super-reflexive Banach spaces, are further main contributions of our
paper. Some of our proofs use main ideas of the article by D. Azagra and C.
Mudarra, but all are formally completely independent on their article.
