Inequality between binomial sums I want to prove that the following inequality holds whenever $k\leq n-1$ and $1\leq i\leq \lfloor\frac{k}{2}\rfloor$.

$$\frac{2}{C}\binom{n}{k}\binom{k}{i+1} \leq \frac{k}{i}\sum_{j=i}^{k-1}
 (k-j)\binom{j-1}{i-1}\binom{n-k+j-1}{j}$$

where $C = \max(k,n-k)+1$.
Two identities I suspect might be very useful here are
$$\binom{k}{i+1} = \sum_{j=i}^{k-1} (k-j)\binom{j-1}{i-1}$$
$$\binom{n-1}{k-1}=\sum_{j=0}^{k-1} \binom{n-k+j-1}{j}$$
I tried to use these and the Chebyshev's inequality (which states that if $a_1\leq \cdots \leq a_n$ and $b_1\leq \cdots \leq b_n$, then $\sum a_i \cdot \sum b_i \leq n \sum a_ib_i$) but it didn't work. I think that this inequality is somehow "tight".
Also, I tried splitting into two cases $k\leq n-k$ and $k\geq n-k$ to have a more concrete expression for $C$. Any help would be highly appreciated.
EDIT: I have tried approaching the case $i=1$, and many identities can be used. The inequality in this case is very tight, and I think that proceeding via induction on $i$ does not help, as the inequality seems to get tighter at every step.
EDIT2: The problem with the Chebyshev approach is that the sequence $(k-j)\binom{j-1}{i-1}$ is not increasing (hence it is not possible to use the Chebyshev's inequality as stated).
EDIT3: I have noticed using Zeilberger's algorithm that
$$\sum_{j=i}^{k-1} j\binom{j-1}{i-1}\binom{n-k+j-1}{j} = \frac{i(i+1)(n-k)}{n(n+i-k)} \binom{n}{k}\binom{k}{i+1}$$
However, no closed expression seems to exist for
$$\sum_{j=i}^{k-1}\binom{j-1}{i-1}\binom{n-k+j-1}{j}$$
 A: (Still a work in progress, see last expression for a sufficient condition on $C$.)
The strategy is to rewrite both sides as multinomials. We then get something that looks similar enough to the hockey stick identity for multinomials.
For the left hand side:
$$\frac{2}{C}\binom{n}{k}\binom{k}{i+1} = \frac{2}{C}\binom{n}{n-k,\,i+1,\,k-i-1}$$
For the right hand side:
$$\frac{k}{i}(k-j)\binom{j-1}{i-1}\binom{n-k+j-1}{j}$$
$$= \binom{n-k+j}{n-k,\, i, \, j-i} \frac{k(k-j)(n-k)}{j(n-k+j)}$$
The hockey stick identity for multinomials (https://www.fq.math.ca/Scanned/34-3/jones.pdf) says that:
$$\binom{n}{n-k,\,i+1,\,k-i-1}$$
$$= \sum_{j=i}^{k-1} \binom{n-k+j}{n-k-1,\, i+1, \, j-i} + \sum_{j=i}^{k-1} \binom{n-k+j}{n-k,\, i, \, j-i}$$
$$= \frac{n-k + i +1}{i+1} \sum_{j=i}^{k-1} \binom{n-k+j}{n-k,\, i, \, j-i}$$
Therefore,
$$\frac{2}{C}\binom{n}{k}\binom{k}{i+1} = \frac{2}{C} \cdot \frac{n-k + i +1}{i+1} \sum_{j=i}^{k-1} \binom{n-k+j}{n-k,\, i, \, j-i}$$
We are done if we can show for all valid $j$ that:
$$\frac{2}{C} \cdot \frac{n-k + i +1}{i+1} \leq \frac{k(k-j)(n-k)}{j(n-k+j)}$$
Or that:
$$\frac{2j(n-k + i +1)(n-k+j)}{k(i+1)(k-j)(n-k)} \leq C$$
Since the left hand side increases in $j$, we can plug in $j=k-1$:
$$\frac{2(k-1)(n-k + i +1)(n-1)}{k(i+1)(n-k)} \leq C$$
We could get tighter bounds by not trying to prove the coefficient is larger for all $j$ but just enough of them (specifically when the multinomial is large). Also, when $k$ is almost $n$, the summation is very flat, and so we don't care as much about $j=k-1$ versus $j=k-2$, etc. So, if $k$ is close to $n$, then the inequality might only need to hold for a somewhat smaller value of $j$. I think, we might need to break up the proof into cases over $k$. (The $k-j$ term in the denominator then becomes pretty important.)
