  Complementarity relations between various characterizations of a probability
distribution are at the core of information theory. In particular, lower and
upper bounds for the entropic function are of great importance. In applied
topics, we often deal with situations, where the sums of certain powers of
probabilities are known. The main question is how to convert the imposed
restrictions into two-sided estimates on the Shannon entropy. It is addressed
in two different ways. The more intuitive of them is based on truncated
expansions of the Taylor type. Another method is based on the use of
coefficients of the shifted Chebyshev polynomials. We propose here a family of
polynomials for estimating the Shannon entropy from below. As a result,
estimates are more uniform in the sense that errors do not become too large in
particular points. The presented method is used for deriving uncertainty and
certainty relations for positive operator-valued measures assigned to a quantum
design. Quantum designs are currently the subject of active researches due to
potential use in quantum information science. It is shown that the derived
estimates are applicable in quantum tomography and detecting steerability of
quantum states.
