  Information entropies give a genuine way to characterize quantitatively an
incompatibility in quantum measurements. Together with the Shannon entropy, few
families of parametrized entropies have found use in various questions. It is
also known that a possibility to vary the parameter can often provide more
restrictions on elements of probability distributions. In quantum information
processing, one often deals with measurements having some special structure.
Quantum designs are currently the subject of active research, whence the aim to
formulate complementarity relations for related measurements occurs. Using
generalized entropies of order $\alpha\in(0,2)$, we obtain uncertainty and
certainty relations for POVMs assigned to a quantum design. The structure of
quantum designs leads to several restrictions on generated probabilities. We
show how to convert these restrictions into two-sided entropic estimates. One
of the used ways is based on truncated expansions of the Taylor type. The
recently found method to get two-sided entropic estimates uses polynomials with
flexible coefficients. We illustrate the utility of this method with respect to
both the R\'{e}nyi and Tsallis entropies. Possible applications of the derived
complementarity relations are briefly discussed.
