Bohr's Inequality Can anyone point me to any reference on the following inequality:
Let $c>0, a,b \in \Bbb{R}$
$$
\|a+b\|^2 \leq (1+c)\|a\|^2+(1+{\textstyle\frac{1}{c}})\|b\|^2
$$
I'm not even sure if "Bohr's Inequality" is this inequality's name, since I can't find it anywhere in the internet.
I'm having a bad day and I got stuck in this one, apparently. Any suggestions or hints on how to prove it will be appreciated.
 A: Do you mean
$$(a+b)^2 \le (1+c)a^2 + (1 + 1/c)b^2$$
?
This follows from $\text{AM} \ge \text{GM}$
$$ca^2 + b^2/c \ge 2 ab$$
No add $a^2 + b^2$ to both sides.
If you meant $\mathbb{R}^n$ and the Euclidean Norm, you can prove it by applying the above $n$ times.
A: Here's a hint. $a,b$ and $c$ are real, and the inequality boils down to showing that
\begin{equation}
c|a|^2 + \frac{1}{c}|b|^2 -2ab \geq 0
\end{equation}
Consider the function
\begin{equation}
f(x) = x|a|^2 + \frac{1}{x}|b|^2 -2ab
\end{equation}
What is the minimum value of this function?
A: This is a remark/comment: 
The inequality you mention is perhaps be an due to Harald Bohr, there is however an other inequality usually called Bohr's inequality. 
Consider a bounded analytic function $f$ on the unit disc, that is $$f(z)=\sum_{n=0}^\infty a_nz^n\ \text{  and  }\ \|f\|_\infty=\sup_{|z|<1}|f(z)|<\infty.$$
Then for $0<r\le\frac{1}{3}$ we have (Bohr's inequality) 
$$\sum_{n=0}^\infty |a_n|r^n\le \|f\|_\infty.$$
Moreover, the constant $\frac{1}{3}$ is the best possible which was actually not proved by Bohr but by F. Wiener. 
H. P. Boas and D. Khavinson, Bohr's power series theorem in several
variables, Proc. Amer. Math. Soc. 125(1997), 2975-2979.
H. Bohr, A theorem concerning power series, Proc. London Math. Soc.
2(13)1914, 1-5.
A: Start with 
\begin{eqnarray*}
\left( a\sqrt{c}-\frac{b}{\sqrt{c} } \right)^2 \geq 0. 
\end{eqnarray*}
Expand, rearrange, add $a^2+b^2$ to both sides and we have 
\begin{eqnarray*}
\left( a+b  \right)^2 \leq (1+c)a^2+ \left( 1+ \frac{1}{c} \right) b^2.  
\end{eqnarray*}
