Hi can you help me with the following
Show that for every $u \ge 0$
$(1 + u)\log(1 + u) - u \ge \cfrac {u^2} {2 + 2u/3}$
Obviously i tried to see something clear from Taylor series of $\log(1+x)$ but didn't see something clear.
Thank you!
Hi can you help me with the following
Show that for every $u \ge 0$
$(1 + u)\log(1 + u) - u \ge \cfrac {u^2} {2 + 2u/3}$
Obviously i tried to see something clear from Taylor series of $\log(1+x)$ but didn't see something clear.
Thank you!
EDIT For the new question, as before consider $f(u) = (1 + u)\log(1 + u) - u - \dfrac{u^2}{2 + 2u/3}$ We then get that $f'(u) = \log(1+u) - \dfrac32 \dfrac{u(u+6)}{(u+3)^2} = \log(1+u) - \dfrac32 + \dfrac{27}2 \dfrac1{(u+3)^2}$ The function $g(x) = \log(1+x) + \dfrac{27}2 \dfrac1{(x+3)^2}$ is an increasing function since $g'(x) = \dfrac1{1+x} - \dfrac{27}{(x+3)^3} = \dfrac{x^2(x + 9)}{(1+x)(x+3)^3} \geq 0$ for all $x \geq 0$. Hence, we get that $f'(u) \geq f'(0) = \log(1) - \dfrac32 + \dfrac{27}2 \dfrac1{3^2} = 0$ Hence, for $u \geq 0$, we have that $f'(u) \geq 0$. Hence, $f(u)$ is increasing for $u \geq 0$. Hence, we get that $f(u) \geq f(0) = 0$ Hence, we can finally conclude that $ (1 + u)\log(1 + u) - u \geq \dfrac{u^2}{2 + 2u/3} $
For the old question. What you have is incorrect. For $u=1/2$, we get that $\dfrac32 \log \left( \dfrac32\right) - \dfrac12 \geq \dfrac{1/2}{2+1/3} = \dfrac3{14}$ i.e., $\dfrac32\log \left( \dfrac32\right) \geq \dfrac3{14} + \dfrac12 = \dfrac57$ i.e., $\log \left( \dfrac32\right) \geq \dfrac{10}{21}$which is incorrect. Your claim is however true for $u>1$ (or) a bit more precisely for $u > 0.971965275483113136037547611917997896698$.
To see this, consider the function $f(u) = (1 + u)\log(1 + u) - u - \dfrac{u}{2 + 2u/3}$ Then $f'(u) = \log(1+u) - \dfrac92 \dfrac1{(u+3)^2}$. The function $g(x) = \log(x) - \dfrac92 \dfrac1{(x+2)^2}$ is an increasing function since $g'(x) = \dfrac1x -\dfrac92 \dfrac{-2}{(x+2)^3} = \dfrac1x + \dfrac9{(x+2)^3} > 0$ for all $x > 0$. Hence, $f'(u) \geq f'(1) \implies \log(1+u) - \dfrac92 \dfrac1{(u+3)^2} \geq \log(2) - \dfrac9{2 \cdot 4^2} = \log(2) - \dfrac9{32} > 0$ Hence, $f'(u) > 0$ for all $u \geq 1$. Hence, $f(u) = \geq f(1) = 2 \log(2) - 1 - \dfrac1{2+2/3} = 2\log 2 - \dfrac{11}8 \approx 0.01129436111 > 0$ Hence for all $u \geq 1$, we get that $(1 + u)\log(1 + u) - u - \dfrac{u}{2 + 2u/3} \geq 0$ i.e. $(1 + u)\log(1 + u) - u \geq \dfrac{u}{2 + 2u/3}$
Reading the book Concentration Inequalities, I found the exact inequality in Ex. 2.8. I solved it using series expansion (of the $\log$ function) and a carefully chosen geometric series thanks to an inequality proved by induction. Both are tricky and less straightforward than the direct approach. I'm posting this for fun and for OP's desire to "see something clear from Taylor series [of the $\log$ function around $1$]".
For any $x>0$, note that $0
$ \begin{aligned} (1+x)\log(1+x)-x =& x \left[ \frac12\left(\frac{x}{1+x}\right) + \frac13\left(\frac{x}{1+x}\right)^2 + \frac14\left(\frac{x}{1+x}\right)^3 + \cdots \right] \\ =& \frac{x^2}{2(1+x)} \left[ 1 + \frac23\left(\frac{x}{1+x}\right) + \frac24\left(\frac{x}{1+x}\right)^2 + \cdots \right] \\ =& \frac{x^2}{2(1+x)} \sum_{k=0}^\infty \frac{2}{k+2} \left(\frac{x}{1+x}\right)^k \end{aligned} $
The following shows that LHS can be bounded below by a geometric series with common ratio $\frac23\left(\frac{x}{1+x}\right),$ so that RHS is shrunk to something simpler than infinite series and more similar to the question.
Lemma For any non-negative integer $k$, we have $\frac{2}{k+2} \ge \left( \frac23 \right)^k.$ Justification by induction
- Base case $k=0$: both sides equal to one
- Induction hypothesis $k=n$: $\frac{2}{n+2} \ge \left( \frac23 \right)^n.$
- Inductive step $k=n+1$: $\frac{2}{n+3} = \frac{2}{n+2} \frac{n+2}{n+3} \ge \left( \frac23 \right)^n \left( 1-\frac{1}{n+3} \right) \ge \left( \frac23 \right)^n \left( 1-\frac13 \right)=\left( \frac23 \right)^{n+1}.\square$
$ \require{cancel} \begin{aligned} (1+x)\log(1+x)-x =& \frac{x^2}{2(1+x)} \sum_{k=0}^\infty \frac{2}{k+2} \left(\frac{x}{1+x}\right)^k \\ \ge& \frac{x^2}{2(1+x)} \sum_{k=0}^\infty \left( \frac23 \right)^k \left(\frac{x}{1+x}\right)^k \\ =& \frac{x^2}{2(1+x)} \frac{1}{1-\frac23\left(\frac{x}{1+x}\right)} \\ =& \frac{x^2}{2\cancel{(1+x)}} \frac{3\cancel{(1+x)}}{3(1+x)-2x} \\ =& \frac{x^2}{2} \frac{3}{x+3} \\ =& \frac{x^2}{2 \left( 1+x/3 \right)} \end{aligned} $