  Operator monotone functions, introduced by Lowner in 1934, are an important
class of real-valued functions. They arise naturally in matrix and operator
theory and have various applications in other branches of mathematics and
related fields. This concept is closely related to operator convex/concave
functions. In this paper, we provide their important examples and
characterizations in terms of matrix of divided differences. Various
characterizations and the relationship between operator monotonicity and
operator convexity are given by Hansen-Pedersen characterizations. Moreover,
operator monotone functions on the nonnegative reals have special properties,
namely, they admit integral representations with respect to suitable Borel
measures.
