[Paper Review] Inequalities for binomial coefficients

Zhi-Hong Sun|arXiv (Cornell University)|Oct 1, 2013

Analytic Number Theory Research4 references3 citations

TL;DR

This paper establishes sharp upper bounds for binomial coefficients ${n \choose k}$ using refined Stirling-type approximations and factorial inequalities. It proves that for $n \geq 400$ and $[n/5] \leq k \leq [n/2]$, ${n \choose k} < \left(1 - \frac{5(k - [n/5])}{6n^2}\right) \frac{n^{n-1/2}}{k^k (n-k)^{n-k}}$, improving upon known asymptotic bounds and confirming a conjecture on extremal binomial coefficient behavior.

ABSTRACT

In this paper we prove several inequalities for binomial coefficients. For instance, if $ k$ and $n$ are positive integers such that $n\ge 400$ and $[\frac n5]\le k\le [\frac n2]$, where $[x]$ is the greatest integer not exceeding $x$, then $$\binom nk

Motivation & Objective

• To refine existing asymptotic bounds for binomial coefficients using tighter factorial approximations.
• To establish explicit, non-asymptotic inequalities for $\binom{n}{k}$ in the range $[n/5] \leq k \leq [n/2]$.
• To confirm a conjecture by a student regarding the dominance of $n^{n-1/2}/(k^k (n-k)^{n-k})$ as an upper bound for $\binom{n}{k}$.
• To investigate the structure of the least $k$ such that $\binom{n}{k} > 2^n / (n+1)$, linking it to extremal binomial coefficient behavior.

Proposed method

• Derives a refined factorial approximation using Hirschhorn's result: $n! = \sqrt{\pi} (n/e)^n (8n^3 + 4n^2 + n + 1/30 - 11/(240n) + r/(240n^2))^{1/6}$ with $5 < r < 11$.
• Applies Robbins' and Feller's bounds on $n!$ to derive inequalities for $\binom{mn}{n}$ and $\binom{n}{k}$.
• Uses logarithmic differentiation and series expansions to bound the ratio $F(n,k) = \frac{\binom{n}{k} k^k (n-k)^{n-k}}{\binom{n}{k+1} (k+1)^{k+1} (n-k-1)^{n-k-1}}$.
• Employs induction and polynomial inequalities to prove $F(n,k) > \exp(11/(12n^2))$ for $k \leq [n/2] - 1$.
• Combines the ratio bounds with the known bound $\binom{n}{k} \leq 2^n$ to derive the final inequality.
• Uses symmetry and monotonicity of $f(k) = k^k (n-k)^{n-k}$ to reduce the analysis to $k \leq n/2$.

Experimental results

Research questions

• RQ1 Can tighter upper bounds be derived for $\binom{n}{k}$ when $k$ is near $n/5$ and $n$ is large?
• RQ2 Is the inequality $\binom{n}{k} < \left(1 - \frac{5(k - [n/5])}{6n^2}\right) \frac{n^{n-1/2}}{k^k (n-k)^{n-k}}$ valid for $n \geq 400$ and $[n/5] \leq k \leq [n/2]$?
• RQ3 What is the exact structure of the least $k$ such that $\binom{n}{k} > 2^n / (n+1)$, and how does it relate to $2^n / n$?
• RQ4 Are there infinitely many $n$ such that $\binom{n}{f(n)} \leq 2^n / n$, where $f(n)$ is the least $k$ with $\binom{n}{k} > 2^n / (n+1)$?

Key findings

• For $n \geq 400$ and $[n/5] \leq k \leq [n/2]$, the inequality ${n \choose k} < \left(1 - \frac{5(k - [n/5])}{6n^2}\right) \frac{n^{n-1/2}}{k^k (n-k)^{n-k}}$ holds.
• The bound improves upon the classical inequality $\binom{n}{k} \leq \frac{n^n}{k^k (n-k)^{n-k}}$ by incorporating a correction factor dependent on $k - [n/5]$.
• The inequality implies $\binom{n}{k} < \left(1 - \frac{5(k - [n/5])}{6n^2}\right) \frac{2^n}{\sqrt{n}}$ for $k \in [n/5, 4n/5]$.
• The paper confirms a conjecture by Shu-Yao Yi that $\binom{n}{k} < \frac{n^{n-1/2}}{k^k (n-k)^{n-k}}$ for $k \in [n/5, 4n/5]$ and $n \geq 400$.
• It is shown that the least $k$ such that $\binom{n}{k} > 2^n / (n+1)$ satisfies $k = f(n)$, and this $k$ is the unique minimizer of $\binom{n}{k}$ in the range $k \leq n/2$ under the given bounds.
• Conjecture 2.1 posits infinitely many $n$ such that $\frac{2^n}{n+1} < \binom{n}{k} \leq \frac{2^n}{n}$, with 18 such pairs found for $n \leq 1500$.