Show that $\left(\frac{x_1^{x_2}}{x_2}\right)^p+\left(\frac{x_2^{x_3}}{x_3}\right)^p+\cdots+\left(\frac{x_n^{x_1}}{x_1}\right)^p\ge n$ for any $p\ge1$

Solution 1:

Proof for $p ≥ 1$

Since $u^p - 1 ≥ p(u - 1)$ for all $u ≥ 0$, it suffices to prove the result for $p = 1$. That follows from

$$\frac{x^y}{y} - 1 ≥ \frac{1 + y \ln x}{y} - 1 = \ln x + \frac1y - 1 ≥ \ln x + \ln \frac1y = \ln x - \ln y$$

by cyclic summation over $(x, y) = (x_i, x_{i + 1})$.

Conjectured proof for $p ≥ \frac12$

Since $u^p - 1 ≥ 2p(u^{\frac12} - 1)$ for all $u ≥ 0$, it suffices to prove the result for $p = \frac12$. Numerical evidence suggests that

$$\left(\frac{x^y}{y}\right)^{\frac12} - 1 ≥ \frac{\ln x}{2\sqrt[4]{1 + \frac13 \ln^2 x}} - \frac{\ln y}{2\sqrt[4]{1 + \frac13 \ln^2 y}}$$

for all $x, y > 0$. If this is true, cyclic summation yields the desired result.

Counterexample for $0 < p < \frac12$

Let $g(x) = \left(\frac{x^{1/x}}{1/x}\right)^p + \left(\frac{(1/x)^x}{x}\right)^p$. Then $g(1) = 2$, $g'(1) = 0$, and $g''(1) = 4p(2p - 1) < 0$, so we have $g(x) < 2$ for $x$ in some neighborhood of $1$. This yields counterexamples for all even $n$:

$$\left(x, \frac1x, x, \frac1x, \dotsc, x, \frac1x\right), \quad x ≈ 1, \quad 0 < p < \frac12.$$

For $n = 3$, the best counterexample seems to be

$$(0.41398215, 0.73186577, 4.77292996), \quad 0 < p < 0.39158477.$$

Solution 2:

We prove the result by first deriving a useful inequality for the function $(x,y)\mapsto (x^y/y)^p$.

It is clear that $$(x^y/y)^p \ge \left(\min_{\alpha \in \mathbb{R}_+} (\alpha x)^{\alpha y}/(\alpha y)\right)^p.\tag{1}$$ For fixed $x$ and $y$, we can find the minimum of the function $\alpha\mapsto (\alpha x)^{\alpha y}/(\alpha y)$ by taking derivatives and setting to $0$. Doing this (e.g., in Mathematica) shows that the minimum is achieved by $\alpha^\star = 1/({yW(ex/y)})$, where $W$ is the Lambert W function. Plugged into $(1)$, this gives $$(x^y/y)^p \ge \left[\left(\frac{x/y}{W(ex/y)}\right)^{1/W(e x/y)}W(ex/y)\right]^p$$ Note that the right hand side is a function of the ratio $x/y$ only. It will be convenient to write it as a function of the logarithm of $x/y$, $(x^y/y)^p\ge f(\ln x/y)$ where $$f(a):=\left[\left(\frac{e^a}{W(e^{a+1})}\right)^{1/W(e^{a+1})} W(e^{a+1})\right]^p.$$

We will use that $f(a)$ is a convex function for all $a\in\mathbb{R}$ and $p\ge 1$, since its second derivative is non-negative. This can be verified in Mathematica by running

FullSimplify[D[((Exp[a]/ProductLog[Exp[a + 1]])^(1/ProductLog[Exp[a + 1]])*ProductLog[Exp[a + 1]])^p, {a, 2}], Assumptions -> {Element[a, Reals]}]

which gives $$f''(a) = \frac{p e^{p-\frac{p}{W\left(e^{a+1}\right)}} W\left(e^{a+1}\right)^{p-2} \left((p-1) W\left(e^{a+1}\right)+p\right)}{W\left(e^{a+1}\right)+1}$$ which is non-negative for $p\ge 1$.

We are now ready to prove the result: $$ \sum_{i=1}^{n} \left(\frac{x_{i}^{x_{i+1}}}{x_{i+1}}\right)^p \ge n,\tag{2} $$ where we use the notation $x_{n+1}=x_1$. To begin, write $$\sum_i \left(\frac{x_{i}^{x_{i+1}}}{x_{i+1}}\right)^p \ge \sum_i f\left(\ln \frac{x_{i}}{x_{i+1}}\right) = n \frac{1}{n} \sum_i f\left(\ln \frac{x_{i}}{x_{i+1}}\right) \ge n f\left(\frac{1}{n} \sum_i \ln \frac{x_{i}}{x_{i+1}}\right)\tag{3}$$ where the first inequality uses $(x^y/y)^p\ge f(\ln x/y)$ while the second inequality is Jensen's. Note that $\sum_i \ln \frac{x_{i}}{x_{i+1}}=0$. It is also easy to verify from inspection that $f(0)=1$. Plugging into $(3)$ gives $$n f\left(\frac{1}{n} \sum_i \ln \frac{x_{i}}{x_{i+1}}\right)= n f(0) = n.$$ Combining with $(3)$ gives the desired result, $(2)$.

Solution 3:

@Artemy's proof involves a complicated part of proving $f''(a)\ge 0$. Here is a proof which is a simplification of @Artemy's proof (one can do it by hand; also avoiding the use of the Lambert W function). Perhaps this is also helpful for the case when $p < 1$ (e.g. $\sqrt{\frac{x^y}{y}}+\sqrt{\frac{y^x}{x}}\ge 2$).

Fact 1: For any given $x, y > 0$, there exists a unique $u > 0$ such that $\frac{x}{y} = u\mathrm{e}^{u - 1}$. Furthermore, $$\frac{x^y}{y} \ge u\mathrm{e}^{1 - 1/u}.$$ (The proof is given at the end.)

Using Fact 1, let $$\frac{x_i}{x_{i + 1}} = u_i\mathrm{e}^{u_i - 1}, \,\, i = 1, 2, \cdots, n - 1; \qquad \frac{x_n}{x_1} = u_n\mathrm{e}^{u_n - 1}$$ where $u_1, \cdots, u_n > 0$. We have \begin{align*} 1 &= \frac{x_1}{x_2}\, \frac{x_2}{x_3} \cdots \frac{x_n}{x_1}\\ &= u_1 u_2 \cdots u_n \mathrm{e}^{u_1 + u_2 + \cdots + u_n - n}\\ &\le \left(\frac{u_1 + u_2 + \cdots + u_n}{n}\right)^n \mathrm{e}^{u_1 + u_2 + \cdots + u_n - n} \end{align*} which results in $$\frac{u_1 + u_2 + \cdots + u_n}{n} \ge 1. \tag{1}$$

Let $$f(u) = (u\mathrm{e}^{1 - 1/u})^p.$$ $f(u)$ is convex on $u > 0$ since $f''(u) = (u\mathrm{e}^{1 - 1/u})^p pu^{-4}[(p - 1)u^2 + 2(p - 1)u + p] > 0$ for all $u > 0$. Also, $f(u)$ is strictly increasing on $u > 0$ since $f'(u) = (u\mathrm{e}^{1 - 1/u})^ppu^{-2}(u + 1) > 0$ for all $u > 0$.

Using Fact 1 and (1), using the convexity and monotonicity of $f(u)$, we have \begin{align*} &\left(\frac{x_1^{x_2}}{x_2}\right)^p + \left(\frac{x_2^{x_3}}{x_3}\right)^p + \cdots + \left(\frac{x_n^{x_1}}{x_1}\right)^p\\ \ge\,& f(u_1) + f(u_2) + \cdots + f(u_n)\\ \ge\,& n\, f\left(\frac{u_1 + u_2 + \cdots + u_n}{n}\right) \\ \ge\,& n f(1)\\ \ge\,& n. \end{align*}

We are done.

Proof of Fact 1:

Since $u\mapsto u\mathrm{e}^{u - 1}$ is strictly increasing on $u > 0$, clearly, there exists a unique $u > 0$ such that $\frac{x}{y} = u\mathrm{e}^{u - 1}$.

We have $$\ln x = \ln y + \ln u + u - 1. $$ We need to prove that $$y\ln x - \ln y \ge \ln u + 1 - \frac{1}{u}.$$ It suffices to prove that $$y(\ln y + \ln u + u - 1) - \ln y \ge \ln u + 1 - \frac{1}{u}.$$

Let $$F(y) = y(\ln y + \ln u + u - 1) - \ln y - \left(\ln u + 1 - \frac{1}{u}\right).$$ We have $$F'(y) = \ln y + \ln u + u - \frac{1}{y},$$ and $$F''(y) = \frac{1}{y} + \frac{1}{y^2} > 0.$$ Also, $F(1/u) = 0$ and $F'(1/u) = 0$. Thus, $F(y) \ge F(1/u) = 0$ for all $y > 0$.

We are done.