On maximizing of the gap in the Jensen inequality and the capacity of quantum channels

The Holevo quantity for a given quantum states ensemble equals by definition the gap in Jensen inequality for von Neumann entropy. We prove a theorem on optimality conditions in the general Jensen gap maximization problem where the function is fixed. Then the theorem derived is applied to the one-shot capacity computation for a certain family of non-unital quantum channels. The results are obtained together with S.N.Filippov.