Does there exist the function $f^2(x)\ge f(x+y)\left(f(x)+y \right) $

Question

Does there exist the function $f:\mathbb R^+\rightarrow \mathbb R^+$, such that $$f^2(x)\ge f(x+y)\left(f(x)+y \right) \forall x,y \in \mathbb R^+$$

My work so far:

Assume that a function exists. Then $$f(x+y)\le \frac{f^2(x)}{f(x)+y}<f(x).$$ Then this function is strictly decreasing. (And this function is injective).

Answer 1

Taking logarithms gives $$ 2\log f(x)\geq\log f(x+y)+\log(1+y/f(x))+\log f(x) $$ After rearranging, $$ \frac{\log f(x)-\log f(x+y)}{y}\geq\frac{\log(1+y/f(x))}{y}, $$ so after taking limit as $y\rightarrow0$ we get $$ \frac{-f'(x)}{f(x)}=(-\log f)'(x)\geq\frac{1}{f(x)}. $$ and so $f'(x)\leq-1$, which contradicts $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$. (As $f$ is decreasing, it is almost everywhere differentiable)

Answer 2

Suppose that there is a function $f$.then we have since $$f^2(x)\ge f(x+y)(f(x)+y)$$ we have $$f(x)-f(x+y)\ge\dfrac{f(x)y}{f(x)+y}>0$$ which shows that $f$is a strictly decreasing function.Give an $x\in R^{+}$, we choose an $n\in N^{+}$ such $nf(x+1)\ge 1$, then $$f\left(x+\dfrac{k}{n}\right)-f\left(x+\dfrac{k+1}{n}\right)\ge\dfrac{f(x+k/n)\cdot\frac{1}{n}}{f(x+\frac{k}{n})+\frac{1}{n}}>\dfrac{1}{2n}$$ summing up these inequalities for $k=0,1,\cdots,n-1$,we have $$f(x)-f(x+1)>\dfrac{1}{2}$$ take an $m$ such $m\ge 2f(x)$,then $$f(x)-f(x+m)>\dfrac{m}{2}\ge f(x)$$a contradiction