  Entropic uncertainty relations are interesting in their own rights as well as
for a lot of applications. Keeping this in mind, we try to make the
corresponding inequalities as tight as possible. The use of parametrized
entropies also allows one to improve relations between various information
measures. Measurements of special types are widely used in quantum information
science. For many of them we can estimate the index of coincidence defined as
the total sum of squared probabilities. Inequalities between entropies and the
index of coincidence form a long-standing direction of researches in classical
information theory. The so-called information diagrams provide a powerful tool
to obtain inequalities of interest. In the literature, results of such a kind
mainly deal with standard information functions linked to the Shannon entropy.
At the same time, generalized information functions have found use in questions
of quantum information theory. In effect, R\'{e}nyi and Tsallis entropies and
related functions are of a separate interest. This paper is devoted to entropic
uncertainty relations derived from information diagrams. The obtained
inequalities are then applied to mutually unbiased bases, symmetric
informationally complete measurements and their generalizations. We also
improve entropic uncertainty relations for quantum measurement assigned to an
equiangular tight frame.
