Lower bound of an inner product

Question

Is there a theorem or equation regarding the lower bound of an inner product?

If it helps, the specific application is the following: I want to show $\sum\limits_\theta \frac{a_\theta}{c_\theta} (a_\theta - b_\theta) \ge 0 $

So $\frac{a_\theta}{c_\theta}$ is the first vector of the inner product and $(a_\theta - b_\theta)$ is the second.

Also: $\sum\limits_\theta a_\theta = 1$

$\sum\limits_\theta b_\theta \le 1$

$\sum\limits_\theta c_\theta \le 1$

$a_\theta, c_\theta > 0$ for all $\theta$

$b_\theta \ge 0$ for all $\theta$.

Thanks in advance!

Answer 1

Let \begin{align} a_1 &= \frac12, & a_2 &= \frac12, \\ b_1 &= 0, & b_2 &= \frac34, \\ c_1 &= \frac12, & c_2 &= \frac1{100}. \\ \end{align} Then $$ \frac{a_1}{c_1} (a_1 - b_1) = \frac12, \qquad \frac{a_2}{c_2} (a_2 - b_2) = -25, $$

$$ \sum\limits_\theta \frac{a_\theta}{c_\theta} (a_\theta - b_\theta) < 0. $$

If $\theta$ was supposed to range over all positive integers, just divide each of the previously-given values of $a_1,a_2,b_1,b_2,c_1,c_1$ by $2$ and set $a_\theta = b_\theta = c_\theta = 2^{1-\theta}$ for $\theta \geq 3$.