# an inequality for multiplication of cubic numbers

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 $c$.

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