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but I`m not indubitably sure. AM-GM doesn`t work. Karamata`s inequality doesn`t work too.

Prove that inequality holds $$ \sum_{k=1}^n (\frac{a_1+\ldots +a_k}{k})^2 \leq 4 \sum_{k=1}^n a_k^2$$

Richard Loo
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  • It is a particular case of Hardy's inequality (https://en.wikipedia.org/wiki/Hardy's_inequality) for $p=2$. Have a look at Pham Kim Hung, Secrets in inequalities. – Jack D'Aurizio May 15 '16 at 19:40
  • ty. but I see some help with proofs

    unfortunately, there is no proof in Jack`s profile

     – Richard Loo May 15 '16 at 19:47
  • Try to use Cauchy-Schwarz inequality in the form $$\frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+\ldots+\frac{a_n^2}{b_n}\geq \frac{(a_1+a_2+\ldots+a_n)^2}{b_1+b_2+\ldots+b_n}.$$ It is a classical inequality. – Jack D'Aurizio May 15 '16 at 19:50
  • http://www.um.es/functanalysis/meetingsold/Zafra/III_Escuela_AF_files/The%20Prehistory%20of%20the%20Hardy%20Inequality.pdf , page 7. – Jack D'Aurizio May 15 '16 at 20:18
  • excuse me, but it is not clear that (8) "obviously" implies (1) for me. could you please explane? – Richard Loo May 15 '16 at 20:23

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