Let
f
f
be a continuous real function on a convex subset of a Banach space. We study what can be said about the semiconcavity (with a general modulus) of
f
f
, if we know that the estimate
\Delta_h^2(f,x) \leq \omega(\|h\|)
Δ
h
2
(
f
,
x
)
≤
ω
(
∥
h
∥
)
holds, where
\Delta_h^2(f,x) = f(x+2h)-2f(x+h) + f(x)
Δ
h
2
(
f
,
x
)
=
f
(
x
+
2
h
)
−
2
f
(
x
+
h
)
+
f
(
x
)
and
\omega:[0,\infty) \to [0,\infty)
ω
:
[
0
,
∞
)
→
[
0
,
∞
)
is a nondecreasing function right continuous at
0
0
with
\omega(0) =0
ω
(
0
)
=
0
. A partial answer to this question was given by P. Cannarsa and C. Sinestrari (2004); we prove versions of their result, which are in a sense best possible. We essentially use methods of A. Marchaud, S. B. Stechkin and others, whose results clarify when the inequality
|\Delta_h^2(f,x)| \leq \omega(\|h\|)
∣
Δ
h
2
(
f
,
x
)
∣
≤
ω
(
∥
h
∥
)
implies that
f
f
is a
C^1
C
1
function (and
f'
f
′
is uniformly continuous with a corresponding modulus of continuity).