Chitkara University Publications

On Some Inequalities Involving Harmonic Mean and Moments


We derive bounds on the second order moment of a random variable in terms of its arithmetic and harmonic means. Both discrete and continuous cases are considered and it is shown that the present bounds provide refinements of the bounds which exist in literature. As an application we obtain a lower bound for the spread of a positive definite matrix A in terms of traces of A, A-1 and A2. Our results compare favourably with those obtained by Wolkowicz and Styan (Bounds for eigenvalues using traces, Lin. Alg. Appl. 29, 471-506, 1980).


  • Madhu gupta, Department of Mathematics, Chitkara University, Solan, India
  • Rajesh Sharma, Department of Mathematics, H.P.U., Shimla, India
  • S.R. Sharma, Department of Mathematics, Chitkara University, Solan, India



Harmonic mean, variance, positive definite matrix, spread


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Merikoski, J.K., On the Trace and Sum of elements of a Matrix, Linear Algebra and Its Applications, 60: 177–185, (1984).

Wolkowicz, H. and Styan, P.H., Bounds for Eigenvalues Using Traces, Linear Algebra and Its Applications, 29: 471–506, (1980).



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