Inequalities involving Weighted Means
If aᵢ > 0, i = 1, 2, …, n and wᵢ ≥ 0, where i = 1, 2, …, n, then weighted arithmetic mean,
\({{A}_{w}}=\frac{{{w}_{1}}{{a}_{1}}+{{w}_{2}}{{a}_{2}}+…+{{w}_{n}}{{a}_{n}}}{{{w}_{1}}+{{w}_{2}}+…….+{{w}_{n}}}\),
Weighted geometric mean,
\({{G}_{w}}={{\left( {{a}_{1}}^{{{w}_{1}}}{{a}_{2}}^{{{w}_{2}}}…{{a}_{n}}^{{{w}_{n}}} \right)}^{\frac{1}{{{w}_{1}}+{{w}_{2}}+…+{{w}_{n}}}}}\),
Weighted harmonic mean,
\({{H}_{w}}=\frac{{{w}_{1}}+{{w}_{2}}+…+{{w}_{n}}}{\frac{{{w}_{1}}}{{{a}_{1}}}+\frac{{{w}_{2}}}{{{a}_{2}}}+…+\frac{{{w}_{n}}}{{{a}_{n}}}}\).
From A.M ≥ G.M ≥ H.M. We have Aw ≥ Gw ≥ Hw
Example: Prove that \({{1}^{1}}\times {{2}^{2}}\times {{3}^{3}}\times ….\times \ nn\ \le \left( \frac{2n+1}{3} \right)\frac{n(n+1)}{2},\ n\in \ N\).
Solution: For 1¹ x 2² x 3³ x … x nⁿ, consider 1, 1 time 2, 2 times …, n, n times
Now
\(\frac{1+\left( 2+2 \right)+\left( 3+3+3 \right)+….+\left( n+n+…+n\ times \right)}{1+2+3+….+n}\,\ge \ {{\left( {{1}^{1}}\times {{2}^{2}}\times {{3}^{3}}\times …..\times {{n}^{n}} \right)}^{\frac{1}{1+2+3+..+n}}}\),
\(\frac{1+{{2}^{2}}+{{3}^{2}}+….+{{n}^{2}}}{\frac{n\left( n+1 \right)}{2}}\ge \ {{\left( {{1}^{1}}\times {{2}^{2}}\times {{3}^{3}}\times …..\times {{n}^{n}} \right)}^{\frac{1}{\frac{n(n+1)}{2}}}}\),
\(\frac{\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}}{\frac{n\left( n+1 \right)}{2}}\ge \ {{\left( {{1}^{1}}\times {{2}^{2}}\times {{3}^{3}}\times …..\times {{n}^{n}} \right)}^{\frac{2}{n(n+1)}}}\),
\(\frac{\left( 2n+1 \right)}{3}\ge \ {{\left( {{1}^{1}}\times {{2}^{2}}\times {{3}^{3}}\times …..\times {{n}^{n}} \right)}^{\frac{2}{n(n+1)}}}\),
\({{\left( \frac{\left( 2n+1 \right)}{3} \right)}^{\left( \frac{n\left( n+1 \right)}{2} \right)}}\ge \ \left( {{1}^{1}}\times {{2}^{2}}\times {{3}^{3}}\times …..\times {{n}^{n}} \right)\).