I am trying set of problems given by my Institute and 1 problem hint asks to derive an inequality but I am unable to do that.
Notations and Inequality to be proved $ j\in \{0, 1,2,...,n\}, n \in\mathbb{N}$ ; $z$ lies on circumference of circle $|z+j+1|= 0.5$ then prove that $|(z+1)_{n+1}|\geq 2^{-3} (j-1)! (n-j-1)! $
Here $(x)_m$ =$(x)(x+1) ... (x+m-1)$.
I tried to use the fact that $z$ lies on circle $|z+j+1|=\frac 12$ and $|a-b|\geq|a|-|b|$. But that doesn't seem to help.
Can someone please derive it.
Considering the issue as a minoration of a product of distances, here is a "geometry based" proof.
Being given $z$, we have a set of translated points $z+1,\cdots z+m$ constrained to belong to translated circles with centers $-j, -j+1, \cdots -j+m-1$ and common radius $1/2$:
Now, in the result
a) the $(1/2)^3$ part deals with the three circles in the center ;
b) the $(j-1)!$ part deals with all the other circles situated on the left ;
c) the $(n-j-1)!$ part deals with all the other circles on the right.
Explanation for a) : Let us call $C_{-1}, C_0, C_{1}$ the circles situated around the origin. The red point on $C_{-1}$ is at least at distance $1/2$ from the origin ; same reasoning for the red point on $C_{1}$ ; the red point on $C_0$ is exactly at distance $1/2$ from the origin ; therefore, the product of these 3 distances is $\ge 1/2^3$.
Explanations for b) and c) are the same, by considering for each circle the least distance case as done upwards.
Remark : an analytic proof is hopefuly possible using this relationship
$$(a)_n=\dfrac{\Gamma(a+n)}{\Gamma(a)}$$
for the "rising factorial" (aka Pochhammer symbol), valid for complex numbers.