Inequalities for combinatorial sums

Abstract

For \(k,l\in \mathbf {N}\), let $$\begin{aligned}&P_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k-1} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }\\&\quad \text{ and }\quad Q_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }. \end{aligned}$$ We prove that the inequality $$\begin{aligned} \frac{1}{4}\le P_{k,l} \end{aligned}$$ is valid for all natural numbers k and l. The sign of equality holds if and only if \(k=l=1\). This complements a result of Vietoris, who showed that $$\begin{aligned} P_{k,l}<\frac{1}{2} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$ An immediate corollary is that $$\begin{aligned} \frac{1}{4}\le P_{k,l}<\frac{1}{2} <Q_{k,l}\le \frac{3}{4} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$ The constant bounds are sharp.