Monotonicity Points and Modulus Estimates in b-Orlicz Function Spaces

Monotonicity points and the modulus of monotonicity are studied in b-Orlicz function spaces over finite nonatomic measure spaces, equipped with the Luxemburg quasi-norm. Lower strictly monotone points and lower and upper locally uniformly monotone points are characterized. For generating functions with constant intervals, upper locally uniformly monotone points may exist even when the Δ2 condition at infinity fails. In this case, all such points are described using the right endpoints of the constant intervals. A criterion for uniform monotonicity is established, including the necessity of the Δα conditions. Lower and upper bounds for the modulus of monotonicity are also established. The exact modulus is computed for nonconvex generating functions given by positive linear combinations of two powers, with one exponent between zero and one and the other above one. For this family, the modulus depends only on the larger exponent. The constants in the general lower estimate are computed for a particular member of this family.