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Question Number 46640 by canhtoan last updated on 29/Oct/18

1≤n,m∈N. Prove that  3(m+n)+10ln (m!n!)≥6(√(mnH_m H_n )).  (H_m =Σ_(i=1) ^m (1/i), H_n =Σ_(j=1) ^n (1/j))

$$\mathrm{1}\leqslant{n},{m}\in\mathbb{N}.\:{Prove}\:{that} \\ $$$$\mathrm{3}\left({m}+{n}\right)+\mathrm{10ln}\:\left({m}!{n}!\right)\geqslant\mathrm{6}\sqrt{{mnH}_{{m}} {H}_{{n}} }. \\ $$$$\left({H}_{{m}} =\underset{{i}=\mathrm{1}} {\overset{{m}} {\sum}}\frac{\mathrm{1}}{{i}},\:{H}_{{n}} =\underset{{j}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{j}}\right) \\ $$

Commented by canhtoan last updated on 29/Oct/18

Help me

$${Help}\:{me} \\ $$

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