Quasiminimizers in one dimension: integrability of the derivative, inverse function and obstacle problems

It is shown that a K-quasiminimizer u for the one-dimensional p-Dirichlet integral is a K'-quasiminimizer for the q-Dirichlet integral, 1 <= q < p(1)(p, K), where p(1)(p, K)> p; the exact value for p(1)(p, K) is obtained. The inverse function of a non-constant u is also K ''-quasiminimizer for the s-Dirichlet integral and the range of the exponent s is specified. Connections between quasiminimizers, superminimizers and solutions to obstacle problems are studied.