On the probability that a discrete IFR random variable exceeds its mean

Let $X$ be an integer-valued random variable with $\mathbb{E}(\mathrm{\mid }X\mathrm{\mid })<\mathrm{\infty }$ whose survival probabilities $S_j=\mathbb{P}(X\ge j),j\in \mathbb{Z}$, form a log-concave sequence. Suppose that for some integer $m$ and some $\gamma \in (0,1\mathrm{/}2]$ it holds $\mathbb{E}(X)\ge m$ and $\mathbb{P}(X=m+1)\ge \gamma \cdot \mathbb{P}(X=m)>0$. It is shown that $$\mathbb{P}(X\ge m+1)\ge \frac{\gamma }{1+2\gamma },$$ with the bound being best possible. As a consequence, it is shown that if $H\sim \text{Hyp}(n,i,k)$ is a hypergeometric random variable for which both $ik\mathrm{/}n$ and $(n-i)(n-k)\mathrm{/}n$ are at least $\log (4\mathrm{/}3)$, then it holds $\mathbb{P}(H>\mathbb{E}(H))\ge 1\mathrm{/}4$ when $\mathbb{E}(H)$ is not an integer, and $\mathbb{P}(H>\mathbb{E}(H))\ge 1\mathrm{/}6$ when $\mathbb{E}(H)$ is an integer, both bounds being best possible.