On upper bounds for the characteristic values of the covariance matrix for multinomial,dirichlet and multivariate hypergeometric distributions

For the characteristic values T1 of the matrix V:=Diag(p)-pp^T with p=(p1,...,pk), p1≥p2≥...≥pk≥pk+1>0 and p1+p2+...+pk+pk+1=1 the inequalities p1≥τ1≥p2≥τ2≥...≥pk≥τk>0 are given by RONNING (1982). These inequalities give, if p and pk+1 are unknown, the upper bound 1≥T1. However, in this note the bound 1/2≥T1 is derived. V is proportional to the covariance matrix for multinomial, Dirichlet and multivariate hypergeometric distributions. A statistical application for the multinomial distribution is given.