Suppose $b > a > 0$, $\delta > 0$.
Let $M_k$ be defined as follows:
For $k < \frac{m}{2}$, $M_k = \frac{{(a + \delta)_k(b)_{m-k} + (a + \delta)_{m-k}(b)_{k} - (a)_{k}(b + \delta)_{m-k} - (a)_{m-k}(b + \delta)_{k}}}{{k!(m - k)!}}$
For $k = \frac{m}{2}$, $M_k = \frac{{(a + \delta)_{k}(b)_{m-k} - (a)_{k}(b + \delta)_{m-k}}}{{k!(m - k)!}}$
where $(a + \delta)_k(b)_{m-k} = r$, $(a + \delta)_{m-k}(b)_{k} = s$, $(a)_{k}(b + \delta)_{m-k} = v$, and $(a)_{m-k}(b + \delta)_{k} = u$.
Note that $v \geq r$, $v \geq s$, $r \geq u$, and $s \geq u$.
Now assume that $M_k \leq 0$ for some $k \leq \frac{m}{2}$, i.e., $r+s\leq v+u$.
We have by inspection $rs \geq uv$ and $v \geq u$. We want to show that the same inequality is true for $k-1$. Nothing but
Prove that:
$r \frac{{b + m - k}}{{a + \delta + k - 1}} + s \frac{{a + \delta + m - k}}{{b + k - 1}} - v \frac{{b + \delta + m - k}}{{a + k - 1}} - u \frac{{a + m - k}}{{b + \delta + k - 1}} \leq 0.$
Now, the author has the following statements to prove the inequality: For $\delta = 0$, we clearly have $\frac{b+m-k}{a+k-1} \geq \frac{a+m-k}{b+k-1},$ so the desired inequality is just a combination of $v \geq r$ and $u + v \geq r + s$ with positive coefficients. Treating $u$, $v$, $r$, $s$ as constants and differentiating the LHS of the above inequality with respect to $\delta$, we get
$ u \frac{{a + m - k}}{{(b + \delta + k - 1)^2}} - v \frac{1}{{a + k -1}} + s \frac{1}{{b + k - 1}} - r \frac{{b + m - k}}{{(a + \delta + k - 1)^2}} $
which is obviously non-positive since $v \geq s$ and $r \geq u$ which proves that $M_{k-1} \leq 0$.
My question is why the author has treated the u, v, r and s as constants. Is he saying that second inequality implies the desired inequality? ( he mentioned that which proves that $M_{k-1} \leq 0$). If so, how does the second inequality imply the desired one? If not, what is the author saying here? Can anyone explain?
Note: I was reading the paper https://doi.org/10.1016/j.jmaa.2009.10.057 [Karp, D. and Sitnik, S.M., 2010. Log-convexity and log-concavity of hypergeometric-like functions. Journal of Mathematical Analysis and Applications, 364(2), pp.384-394]. One can see this in section 2 General Theorems, Theorem 1.