×

Refined arithmetic, geometric and harmonic mean inequalities. (English) Zbl 1066.26020

Noting that \(1-1/x\) is a concave function the author applies the Hermite-Hadamard inequality \[ {f(a) + f(b)\over 2}\leq {1\over b-a}\int_a^b f\leq f\bigl((a+b)/2\bigr) \] to obtain the inequalities \[ {(x-1)^2\over2x}\leq(\geq)\, x-1-\log x\leq(\geq)\, {(x-1)^2\over x+1},\quad x>1(\leq 1). \] Substituing \(a_i/G, a_i/A, H/a_i\) (here \(A,G,H\) are the arithmetic, geometric and harmonic means), for \(x\), mutiplying by \(w_i \) and summing over \(i\) leads to interesting estimates for \(A-G, \log A/G\), and \(\log G/H\). Starting with the concave function \(1-1/x^2\) a similar argument gives estimates for \( A-H\). An application is made to further improve the Ky Fan inequality.

MSC:

26D15 Inequalities for sums, series and integrals
26E60 Means
PDFBibTeX XMLCite
Full Text: DOI