A new proof of the Benedetti's inequality and some applications to perturbation to real eigenvalues and singular values

Using the standard deviation of the real samples μ_n ≥ … ≥ μ₁ and λ_n ≥ … ≥ λ₁, we refine the Chebyshev's inequality (refer to [5]),

As a consequence, we obtain a new proof of the Benedetti's inequality (refer to [1], [2] and [4])

where Cov[μ, λ], s(μ) and s(λ) denote the covariance, and the standard deviations (≠ 0) of the sample vectors μ = (μ₁, …, μ_n) and λ = (λ₁, …, λ_n), respectively.
We can also get very interesting applications to eigenvalues and singular values perturbation theory. For some kinds of matrices, the result that we present improves the well known Homand-Weiland's inequality.