Let $f: R \rightarrow R$. Consider the following properties:

$(1)$ - There are positive constants $a$ and $r$ such that $\forall x, y$ $$|f(x)-f(y)|\leq a(1 + |x|^r+|y|^r)|x-y|.$$

$(2)$ - There is a positive constants $b$ such that $\forall x, y$ $$(f(x)-f(y))(x-y)\leq b(x-y)^2.$$

$(3)$ - $f(0)=0$, (or $f(0)$ is limited).

$(4)$ - There is a positive constant $c$ such that $$f(x)x \leq c(1 + x^2).$$

The questions are:

$(1) + (3) + (4) \Rightarrow (2)$ ????

or

There is $f$ that satisfies $(1) + (3) + (4)$ but, not satisfies $(2)$????

REMARK: Is easy to see that $(1) + (2) + (3)$ implies $(4)$, or only, $(2) + (3)$ implies $(4)$.

`$f(x)=\begin{cases}x^{42}(1+\sin x),&x<0,\\0,&x\ge0\end{cases}$`

is a counterexample. Are youreallysure you don’t want absolute values on the left-hand side of (4)? – Emil Jeřábek Aug 12 '11 at 17:11