I want to know when the following statement is true
$$ f(i)=\binom{N-i+1}{i} \leq \binom{N-(k-i)+1}{k-i} = g(i), $$ where $0 \leq i \leq k$ and $0 \leq k \leq N$ and $N, k$ are chosen beforehand and $i $ varies.
I've got the following idea, when $i = k/2$ we clearly have equality. We have a symmetry of the equations that $f(k/2 - i) = g(k/2 + i)$. From plotting in Mathematica I get the idea that for $i \leq k/2$ the inequality is indeed true, but proving it seems quite difficult.
Are there any Binomial identities one can use for this?
The inequality is false, for instance, for $N=4$ and $k=i=2$. Then $f(i)=3$, but $g(i)=1$.