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The potential interaction λx2/(1+gx2), g>0, of the harmonic oscillator H0=−d2/dx2+x2 considered as an operator in the space L2(−∞, ∞) is bounded. This together with the nondegeneracy of the eigenvalues implies that the eigenvectors of the perturbed harmonic oscillator as functions of the parameters λ and g are strongly differentiable. The eigenvalues are therefore differentiable functions for every real λ and every real g>0. In particular, the first eigenvalue E1(λ) as a function of λ is strictly concave (E″1(λ)<0). This paper, exploiting the above properties, aims at several inequalities for the eigenvalues of H0+λx2/(1+gx2), g>0. Emphasis is given to the inequality that follows from the strict concavity of the function E1(λ).