  Inequalities are key tools to prove FDR control of a multiple test. The
present paper studies upper and lower bounds for the FDR under various
dependence structures of p-values, namely independence, reverse martingale
dependence and positive regression dependence on the subset (PRDS) of true null
hypotheses. The inequalities are based on exact finite sample formulas which
are also of interest for independent uniformly distributed p-values under the
null. As applications the asymptotic worst case FDR of step up and step down
tests coming from an non-decreasing rejection curve is established. In
addition, new step up tests are established and necessary conditions for the
FDR control are discussed. The reverse martingale models yield sharper FDR
results than the PRDS models. Already in certain multivariate normal dependence
models the familywise error rate of the Benjamini Hochberg step up test can be
different from the desired level alpha. The second part of the paper is devoted
to adaptive step up tests under dependence. The well-known Storey estimator is
modified so that the corresponding step up test has finite sample control for
various block wise dependent p-values. These results may be applied to
dependent genome data. Within each chromosome the p-values may be reverse
martingale dependent while the chromosomes are independent.
