On one extreme value problem for entropy and error probability

The problem of determining both the maximum and minimum entropy of a random variable $Y$ as well as the maximum absolute value of the difference between entropies of $Y$ and another random variable $X$ is considered under the condition that the probability distribution of $X$ is fixed and the error probability (i.e., the probability of noncoincidence of random values of $X$ and $Y$) is given. A precise expression for the minimum entropy of $Y$ is found. Some conditions under which the entropy of $Y$ takes its maximum value are pointed out. In other cases, some lower and upper bounds are obtained for the maximum entropy of $Y$ as well as for the maximum absolute value of the difference between entropies of $Y$ and $X$.