I would like to ask about a certain inequality that I need and which came out of some [work in here](https://www.math.temple.edu/~tewodros/Catalan_Convexity.pdf).



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> **Question.** For integers $n\geq1$ and $k\geq3$, is this true? If so, any proof?
>  $$6\prod\_{j=1}^k(3n+j)\geq k!\,(nk+3)(nk+2)(nk+1).$$
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*Caveat.* I'm not particularly interested in asymptotic analysis because for fixed $n$ it is rather clear what happens when $k$ is large.



A two-parameter inequality on product of linear terms
For $n=1$ we have an equality, so assume that $n\geqslant 2$. For $k=3$ we have an equality, so induct on $k$ proving that $LHS/RHS$ increases if we replace $k$ to $k+1$. This is equivalent $$1+\frac{3n}{k+1}\geqslant \left(1+\frac{n}{nk+3}\right)\left(1+\frac{n}{nk+2}\right)\left(1+\frac{n}{nk+1}\right).$$
Each multiple on the right does not exceed $1+1/k$, thus it suffices to prove that $1+6/(k+1)\geqslant (1+1/k)^3$, that is true for $k\geqslant 3$. 


