Prove $\mathbb{P}( k < l/2 ) \geq \frac{l}{2} \times \mathbb{P}( k = l/4 ) $ for binomial variable $k$ Suppose we have a Binomial variable: 
$$
k \sim Bin(l,\alpha)
$$
Is it possible to prove/disprove that: 
$$
\mathbb{P}( k < l/2 ) \geq \frac{l}{2} \times \mathbb{P}( k = l/4 )  
$$
EDIT: it's been used in 2nd line of 3rd paragraph in http://arxiv.org/pdf/1110.3564v4.pdf 
(page 33)

As stated in the comments, the inequality holds vacuously if $l$ is not a multiple of four.
Assuming $l=4n$, if we prove that the pdf of $\operatorname{Bin}(\alpha,4n)$ is convex on the interval $[0,2n]$, then the inequality follows from Jensen's inequality. Notice that the inflection points of the pdf of the standard normal distribution occurs in $x=\pm 1$, so it is reasonable to expect that the pdf of $\operatorname{Bin}(\alpha,4n)$ is convex on the set $|x-\mu|\geq\sigma$, or
$$ |x-4n\alpha|\geq\sqrt{4n\alpha(1-\alpha)},$$
so if $\alpha\geq\frac{1}{2}$ we're OK. This is exactly the assumption in the linked paper.
