  In this paper, we discuss the worst-case of distortion riskmetrics for
general distributions when only partial information (mean and variance) is
known. This result is applicable to general class of distortion risk measures
and variability measures. Furthermore, we also consider worst-case of weighted
entropy for general distributions when only partial information is available.
Specifically, we provide some applications for entropies, weighted entropies
and risk measures. The commonly used entropies include Gini functional,
cumulative residual entropy, tail-Gini functional, cumulative Tsallis past
entropy, extended Gini coefficient and so on. The risk measures contain some
premium principles and shortfalls based on entropy. The shortfalls include the
Gini shortfall, extended Gini shortfall, shortfall of cumulative residual
entropy and shortfall of cumulative residual Tsallis entropy with order
$\alpha$.
