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Let $a_1,$ $a_2,$ $\ldots,$ $a_n$ be positive real numbers. Prove that $$\sqrt{\frac{a_1^2+\left( \frac{a_1+a_2}{2}\right)^2+\cdots +\left(\frac{a_1+a_2+\cdots +a_n}{n}\right)^2}{n}} \le \frac{a_1+\sqrt{\frac{a_1^2+a_2^2}{2}}+\cdots+\sqrt{\frac{a_1^2+a_2^2+\cdots +a_n^2}{n}}}{n}.$$ I have proved this inequality for $n=2$ and $n=3.$ But I still cannot prove it for the general case. Can somone help me? |
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Mixed mean inequalities have been studied quite a bit, inspired mostly by the inequalities of Carleman and Hardy, starting probably from this article of K. Kedlaya. Note that the following holds if $r < s$ (in your case $r=1, s=2$): $$\left(\frac{1}{n}\sum_{k=1}^n \left(M_k^{[r]}(\mathbf a)\right) ^s\right)^{1/s}\leq \left(\frac{1}{n}\sum_{k=1}^n\left(M_k^{[s]}(\mathbf a)\right) ^r \right)^{1/r}$$ Where $\mathbf a$ denotes a sequence of real numbers, and $M_k^{[r]}$ denotes the $r$-th power mean of the first $k$ variables. You can find a proof of the general form in "Survey on classical inequalities" by T.M. Rassias (page 32, it is the original proof of B. Mond and J. Pecaric which was later extended to matrices and linear operators). |
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