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Define the function $f:[0,1) \rightarrow [0,\infty) $ given by $$ \large{f(\theta)} := (1-\theta^2)^p (1+\theta)^{p \theta} (1-\theta)^{-p \theta}.$$

If $p$ is a sufficiently large positive real number, is it true that $$ \large{f(\theta) \geq C(1+p \theta^2)}$$ for some positive constant $C$? To clarify, the constant $C$ is independent of $p$ and $\theta$?

Note that upto second order $f(\theta)$ is equal to $(1+p\theta^2)$. To see why, note that $$ (1-\theta ^2)^p = 1- p \theta ^2 + O(\theta^3), \qquad (1+\theta)^{p \theta} = 1+ p \theta^2 + O(\theta^3), \qquad (1-\theta)^{-p \theta} = 1+ p \theta^2 + O(\theta^3). $$

This proves that upto second order $f(\theta)$ is equal to $(1+p\theta^2)$.

Remark: If one can show $$(1+\theta)^{1+\theta} (1-\theta)^{1-\theta} \geq 1+\theta^2,$$ then we are done, via Bernouli's inequality. This was suggested by someone, in a now deleted post. This inequality does seem to be true (I tried plugging in $\theta = 0.1, 0.2, \ldots, 0.9$....it holds true). Note that this inequality doesn't depend on $p$, hence its probably more tractable.

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    $\begingroup$ Are you sure about the last sentence ? $\endgroup$ Nov 17 '14 at 11:29
  • $\begingroup$ @Claude: Sorry, I made a mistake of a minus sign. I believe the last statement is now correct (I have also changed the inequality I want as a result). $\endgroup$
    – Ritwik
    Nov 17 '14 at 11:41
  • $\begingroup$ Much better, indeed ! $\endgroup$ Nov 17 '14 at 11:45
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For your last inequality(in your remark), let $x\in (0,1)$, you have::

$$(1+x)\log(1+x)=x+\sum_{n\geq 2}\frac{(-1)^n}{n(n-1)}x^n$$

Hence $$(1+x)\log(1+x)+(1-x)\log(1-x)=\sum_{n\geq 2, 2|n}\frac{2x^n}{n(n-1)} \geq x^2$$ Thus: $$(1+x)^{1+x}(1-x)^{1-x}\geq \exp(x^2)\geq 1+x^2$$

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  • $\begingroup$ Fantastic! This is very neat and elegant! $\endgroup$
    – Ritwik
    Nov 17 '14 at 14:23
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Since $$\frac{d^2}{dx^2} \ln\left(\frac{1+x}{1-x}\right) = \frac{4x}{(1-x^2)^2} $$ we have that $\ln\bigl(\frac{1+x}{1-x}\bigr)$ is convex on $[0,1)$. So, if $x\in[0,1)$ and $\lambda\in[0,1]$, then $$ \ln\left(\frac{1+((1-\lambda)0 + \lambda x)} {1-((1-\lambda)0 + \lambda x)}\right) \le (1-\lambda)\ln\left(\frac{1+0}{1-0}\right) +\lambda \ln\left(\frac{1+x}{1-x}\right) $$ Simplifying, we get $$ \frac{1+\lambda x}{1-\lambda x} \le \left(\frac{1+x}{1-x}\right)^\lambda \qquad\text{if $x\in[0,1)$ and $\lambda\in[0,1]$.} \tag1 $$ (This is reminiscent of Bernoulli's inequality, but doesn't seem to imply it or be implied by it.) In particular, taking $\lambda=x\in[0,1)$, we get $$ \frac{1+x^2}{1-x^2} \le \left(\frac{1+x}{1-x}\right)^x \qquad\text{if $x\in[0,1)$.} $$ Multiplying by $1-x^2$ yields the desired inequality $$ 1+x^2 \le (1+x)^{1+x}(1-x)^{1-x} $$


Remark: the convexity of $g(x)=\ln\bigl(\frac{1+x}{1-x}\bigr)$ can also be established without calculus. Indeed, if $a>b$ then $t\mapsto\frac{a+t}{b+t}$ is a decreasing function of $t$; since $\bigl(\frac{x+y}2\bigr)^2\ge xy$ by AM/GM, this yields $$ \frac{1+x+y+\tfrac14(x+y)^2}{1-x-y+\tfrac14(x+y)^2} \le \frac{1+x+y+xy}{1-x-y+xy} $$ That is, $$ \frac{\bigl(1+\tfrac12(x+y)\bigr)^2}{\bigl(1-\tfrac12(x+y)\bigr)^2} \le \frac{(1+x)(1+y)}{(1-x)(1-y)} $$ Taking square roots, then logs, yields $g(\frac12(x+y))\le\frac12(g(x)+g(y))$, which since $g$ is continuous establishes that $g$ is convex.


An additional remark: If $a$ and $b$ are positive then $$ \frac{a+b}2 \le \sqrt{\frac{a^2+b^2}{2}} \le a^{a/(a+b)} b^{b/(a+b)} $$ The first inequality is the standard QM/AM inequality. The fact that the far left is less than the far right is an instance of the standard GM/HM inequality. The second inequality is a variant of yours: take $x=\frac{a-b}{a+b}$. In this form, your inequality fits very nicely into the general scheme of standard mean inequalities.

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  • $\begingroup$ The answer by @Kelenner is quite a bit nicer, but I thought this was an interesting alternative. $\endgroup$
    – user21467
    Nov 17 '14 at 15:00
  • $\begingroup$ Yes, this is also a nice solution. $\endgroup$
    – Ritwik
    Nov 17 '14 at 15:13

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