exponential inequality $\frac{2^{2C_1}+2^{2C_2}-1}{2^{2C_1}}\geq\frac{2^{2\hat{C_1}}+2^{2\hat{C_2}}-1}{2^{2\hat{C_1}}}$

Question

I am trying to see if this inequality is correct or not:

$\frac{2^{2C_1}+2^{2C_2}-1}{2^{2C_1}}\geq\frac{2^{2\hat{C_1}}+2^{2\hat{C_2}}-1}{2^{2\hat{C_1}}}$

Given that: $C_1\geq C_2$, $\hat{C_1}\geq \hat{C_2}$, $C_1\geq\hat{C_1}$,$C_2\geq\hat{C_2}$.

My attempt:

$2^{2C_1+2\hat{C_1}}+2^{2C_2+2\hat{C_1}}-2^{2\hat{C_1}}\geq 2^{2\hat{C_1}+2C_1}+2^{2\hat{C_2}+2C_1}-2^{2C_1}$

$2^{2C_2+2\hat{C_1}}-2^{2\hat{C_1}}\geq 2^{2\hat{C_2}+2C_1}-2^{2C_1}$

$2^{2C_1}+2^{2C_2+2\hat{C_1}}\geq 2^{2\hat{C_1}}+2^{2\hat{C_2}+2C_1}$

Answer 1

Consider your last line $$ 2^{2C_1}+2^{2C_2+2\hat{C_1}}\geq 2^{2\hat{C_1}}+2^{2\hat{C_2}+2C_1} $$ Select $C_1 = 5$, $\hat{C_1} = 2$, $C_2 = 2$,$\hat{C_2} = 1$.

This satisfies the given conditions: $C_1\geq C_2$, $\hat{C_1}\geq \hat{C_2}$, $C_1\geq\hat{C_1}$,$C_2\geq\hat{C_2}$.

Plugging this into the eq: $$ 2^{2\cdot 5}+2^{2\cdot 2+2 \cdot 2}\geq 2^{2\cdot 2}+2^{2\cdot 1+2 \cdot 5} $$ we see that this is violated since the numbers are $$ 2^{2\cdot 5}+2^{2\cdot 2+2 \cdot 2} = 1280\geq 2^{2\cdot 2}+2^{2\cdot 1+2 \cdot 5} = 4112 $$ So the inequality does not hold.

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This didn't materialize out of thin air, here is some more explanation how I "made this up":

With some real $a$, multiply both sides of your last line with $2^{4 a}$ which doesn't change the inequality. You get $$ 2^{2(C_1+2a)}+2^{2(C_2+a)+2(\hat{C_1} +a)}\geq 2^{2(\hat{C_1}+2a)}+2^{2(\hat{C_2}+a)+2(C_1+a)} $$ Shifting all variables by $a$ also doesn't change the given conditions. So in the shifted variables, you have $$ 2^{2a +2C_1}+2^{2C_2+2\hat{C_1}}\geq 2^{2a +2\hat{C_1}}+2^{2\hat{C_2}+2C_1} $$ which gives that
$$ 2^{2a}\geq \frac{2^{2\hat{C_2}+2C_1} - 2^{2C_2+2\hat{C_1}}}{2^{2{C_1}} - 2^{2\hat{C_1}}} $$ is required. Now you can choose some values on the RHS and then compute an $a$ which actually violates this.

I chose $C_1 = 4$, $\hat{C_1} = 1$, $C_2 = 1$,$\hat{C_2} = 0$ - remember these are the shifted variables. Then $$ 2^{2a}\geq \frac{2^{8} - 2^{4}}{2^{8} - 2^{2}} \simeq 0.95 $$ or $a > -0.035$. So this can be violated by, say, $a = -1$. Then we have the original values $C_1 = 5$, $\hat{C_1} = 2$, $C_2 = 2$,$\hat{C_2} = 1$ which are used above.

Answer 2

The inequality is not true in general, indeed we have that

$$\frac{2^{2C_1} +2^{2C_2} -1}{2^{2C_1}}\geq\frac{2^{2\hat{C_1}}+2^{2\hat{C_2}}-1}{2^{2\hat{C_1}}}\iff 1 +4^{C_2-C_1}-4^{-C_1}\ge1 +4^{\hat C_2-\hat C_1}-4^{-\hat C_1}$$ $$\iff 4^{C_2-C_1}-4^{-C_1}\ge 4^{\hat C_2-\hat C_1}-4^{-\hat C_1}$$ $$\iff 4^{-C_1}(4^{C_2}-1)\ge 4^{-\hat C_1}(4^{\hat C_2}-1)$$

now let consider

$$C_2=\hat C_2=\hat C_1>0\implies 4^{-C_1} (4^{C_2}-1)\ge 4^{-C_2}(4^{C_2}-1)\implies 4^{-C_1}\ge 4^{-C_2}$$

which is not true since $4^{-x}$ is decreasing and $C_1\ge C_2$.