According to a celebrated result by Löwner, a real-valued function
f
f
is operator monotone if and only if its Löwner matrix, which is the matrix of divided differences
L
f
=
(
f
(
x
i
)
−
f
(
x
j
)
x
i
−
x
j
)
i
,
j
=
1
N
L_f=\left (\frac {f(x_i)-f(x_j)}{x_i-x_j}\right )_{i,j=1}^N
, is positive semidefinite for every integer
N
>
0
N>0
and any choice of
x
1
,
x
2
,
…
,
x
N
x_1,x_2,\ldots ,x_N
. In this paper we answer a question of R. Bhatia, who asked for a characterisation of real-valued functions
g
g
defined on
(
0
,
+
∞
)
(0,+\infty )
for which the matrix of divided sums
K
g
=
(
g
(
x
i
)
+
g
(
x
j
)
x
i
+
x
j
)
i
,
j
=
1
N
K_g=\left (\frac {g(x_i)+g(x_j)}{x_i+x_j}\right )_{i,j=1}^N
, which we call its anti-Löwner matrix, is positive semidefinite for every integer
N
>
0
N>0
and any choice of
x
1
,
x
2
,
…
,
x
N
∈
(
0
,
+
∞
)
x_1,x_2,\ldots ,x_N\in (0,+\infty )
. Such functions, which we call anti-Löwner functions, have applications in the theory of Lyapunov-type equations.