Consider :
- $\alpha$ , $\lambda$ and $\rho$ $\in \mathbb{R}$
- $\alpha$ $\geq$ - $\frac{1}{2}$
- $a=\alpha$ + $\frac{1}{2}$ and $b=\frac{\lambda^2}{2(\alpha+1)}$
a function $f(x)\in \mathbb{R}$ , $x>0$
$f$ is decreasing to $2\rho$
ie: $f$ is decreasing and $$\lim_{x\rightarrow\infty} f(x) = 2\rho\; \forall x > 0.$$
Consider a function $H:$ $$H(x) = \frac{a}{x} - \frac{1}{2}f(x) - b\ , \forall x > 0.$$
I need to know:
- in which cases $$\exists\ c \in \mathbb{R}\ : H(x) < c\ , \forall x > 0.$$
- in these cases what are the values of $c$ ?
Thanks.
Update
$$f(x) = \frac{2 \alpha + 1}{x} + \beta(x)\ , \forall x > 0.$$
where $\beta$ is a function and $\beta(0)=0$