Let \(q \ge 2\) be an integer and let \(p\) be a real number with \(0 \lt p \lt 1\). Then the \(q\)-th power of the \(q\)-ary entropy \(H_q(p)\) is given by

Let \(q \ge 2\) be an integer and let \(p\) be a real number with \(0 \lt p \lt 1\). Then the base-\(q\) power of the \(q\)-ary entropy \(H_q(p)\) admits the closed form

For every real number \(x \ge 0\),

where \(\lfloor x \rfloor \) denotes the greatest integer not exceeding \(x\).

Fix a real number \(p\) with \(0 \lt p \lt 1\) and an integer \(q \ge 2\), and let \(H_q\) denote the \(q\)-ary entropy function. Then there is a function \(\varepsilon \colon \mathbb {N}\to \mathbb {R}\) with \(\varepsilon (n) = o(n)\) such that, for all sufficiently large \(n\),

For every natural number \(q\) and every real number \(p\),

where \(\log _q\) denotes the base-\(q\) logarithm. That is, the three terms of the \(q\)-ary entropy expression may be rewritten by commuting each product and absorbing the subtractions into the sign of the multiplier.

For any nonnegative real numbers \(a, b \ge 0\), the geometric mean is at most the arithmetic mean:

For every real number \(n \ge 0\), the three square roots \(\sqrt{2\pi n}\), \(\sqrt{\pi /2}\), and \(\sqrt{n}\) multiply to \(\pi n\); that is,

Let \(p\) be a real number with \(0 \lt p\) and \(0 \lt 1 - p\), and set \(N_2 = \lceil 2/(p(1-p))\rceil + 1\). Then for every natural number \(n\) with \(N_2 \le n\), the floor \(\lfloor np\rfloor \) is strictly positive, i.e. \(\lfloor np\rfloor \ge 1\).

Let \(a,b,n\in \mathbb {N}\) with \(a\gt 0\), \(b\gt 0\), and \(a+b=n\), and let \(c\gt 0\) be a real number. Suppose the factorial product \(a!\, b!\) admits the Stirling-type upper bound

Then the ratio \(n!/(a!\, b!)\) is bounded below by