Minimum of a real-valued function of multiple variables

Suppose we are given two functions $$\phi(x) = \frac{\sigma^2}{2\mu^2} \left(1 - e^{-(2\mu/\sigma^2)x}\right) - \frac{x}{\mu}，x\in \mathbb R$$ and $$g_0(x) = \frac{\sigma^6}{4\mu^4} \left(1-e^{-(2\mu/\sigma^2)x}\right) - \frac{1}{3\mu}x^3 + \frac{\sigma^2}{2\mu^2} x^2 - \frac{\sigma^4}{2\mu^3} x，x\in \mathbb R$$ where $\mu$ and $\sigma$ are positive constants. Let also $a$ and $b$ be positive constants. Does there exist a real number, denoted by $m$, such that the following inequality holds? $$m (\phi(z) -\phi(y) ) \le a + b |y-z| + g_0(z) -g_0(y), \text{ for all }(y,z) \in \mathbb R^2.$$ Further, if the answer is yes, can we find two pairs $(\hat w,\hat x)$ and $(\hat y,\hat z)$ with $\hat w < \hat x$ and $\hat y>\hat z$ such that $$F(\hat w, \hat x)=F(\hat y,\hat z) =m,$$ where the function $F$ is defined as $$F(y,z):=\frac{a + b |y-z| + g_0(z) -g_0(y)}{\phi(z) -\phi(y)}?$$