The efficiency of secret sharing schemes is measured by means of the average or worst-case information rate. It has been shown that information theoretical inequalities can be used to upper bound these rates. Recently, Jackson and Martin studied the optimal average and worst-case information rate of access structures on five participants. Using combinatorial arguments they showed that information theoretically derived upper bounds for certain optimal average information rates can never be met with equality. This led to the search of new information theoretical inequalities to be used in secret sharing. In this letter we discuss one example from Jackson and Martins' paper. We improve the optimal average information rate by introducing new inequalities.