I need to know is there a positive constant $c$ such that $$ |a_1 \cdots a_n|^3 \leq c\big(|a_1|^3+\cdots+|a_n|^3\big), $$ where $a_{i}\neq 0$ ? I tried geometric and arithmetic inequalities but I did not find such a $c$.

  • $\begingroup$ Try to put $a_i=2$ for all $i$ and see where that gets you … $\endgroup$ – Harald Hanche-Olsen Jun 27 '15 at 21:20
  • $\begingroup$ Try $\forall i, a_i=2t$ and see what happens as $t\to+\infty$. $\endgroup$ – user228113 Jun 27 '15 at 21:27
  • $\begingroup$ LHS degree $3n$, RHS degree $3$, no way $\endgroup$ – Orest Bucicovschi Jun 27 '15 at 21:31

As others have pointed out, this inequality is false for any value of $c$. Just take $a_1=a_2=\cdots=a_n=x>1$: then the inequality would imply $$ x^{3n}\leq cn x^3. $$ The left side grows exponentially in $n$, so it is much larger than the right hand side when $n\to\infty$.


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