Reduction to particular cases of a lemma 
Lemma:Let $\phi(s)$ be a non-negative and non-decreasing function. Suposse that 
  \begin{equation}
\phi(r) \le C_1 \left[\Bigl(\dfrac{r}{R}\Bigr)^\alpha + \mu \right]\phi(R) + C_2 R^\beta]
\end{equation}
  for all $r\le R \le R_0$, with $C_1,\alpha,\beta$ positive constants. Then, for any $\sigma<\min\{\alpha,\beta\}$ there exists a constant $\mu_0 = \mu_0(C_1,\alpha,\beta,\sigma)$ such that if $\mu<\mu_0$, then for for all $r\le R\le R_0$ we have
  \begin{equation}
\phi(r)\le C_3\Bigl(\dfrac{r}{R}\Bigr)^\sigma \left[\phi(R) + C_2R^\sigma \right]
\end{equation}
  where $C_3=C_3(C_1,\sigma-\min\{\alpha,\beta\})$ is a positive constant. In turn, 
  \begin{equation}
\phi(r)\le C_4r^\sigma,
\end{equation}
  where $C_4=C_4(C_2,C_3,R_0,\phi,\sigma)$ is a positive constant.

proof:
We can assume $\beta < \alpha$ and, in this case, it suffices to show the estimate for $\sigma = \beta. \cdots$
Ask I'd like to prove the  assertion in the first line of the proof above.
The lemma can be found here on page 9 and a similar lemma can be found in article,book in the end on page 10.
 A: The first line says two things : 
(1) We may assume $\beta < \alpha$ in (2.6)
and
(2) In this case, we may further assume that $\sigma=\beta$.
Note that (2.6) still holds if we replace $\alpha$ by a larger value $\alpha’ > \alpha$ , because
$$
\Bigl(\dfrac{r}{R}\Bigr)^\alpha \le \Bigl(\dfrac{r}{R}\Bigr)^{\alpha’}
$$ 
So we may enlarge $\alpha$ as we please, and in particular we can make it $> \beta$. This justifies (1).
(2.7) can be written as $\phi (r) \leq F(C_2,C_3,\sigma,r,R)$ where $F$ is the complicated function defined by
$$
F(C_2,C_3,\sigma,r,R)=C_3\Bigl(\dfrac{r}{R}\Bigr)^\sigma \left[\phi(R) + C_2R^\sigma \right]
$$
Suppose that we have shown the estimate (2.7) when $\sigma=\beta$, so that $\phi (r)  \leq F(C_2,C_3,\beta,r,R)$ for $r \leq R \leq R_0$. Now let
$$
C’_3={\sf max}\Bigg(C_3,C_3R^{\beta-\sigma}\Bigg)
$$
This is a positive constant, and we have for $r \leq R$,
$$
C_3\Bigl(\dfrac{r}{R}\Bigr)^\sigma \leq C_3 \leq C’_3 \ \ {\text{and}}  \ \ C_3r^{\beta-\sigma} \leq C_3R^{\beta-\sigma} \leq C’_3
$$
and hence,
$$
C_3\Bigl(\dfrac{r}{R}\Bigr)^\beta \phi (R) \leq C’_3\Bigl(\dfrac{r}{R}\Bigr)^\sigma \phi (R)
 \ \ {\text{and}}  \ \ C_2C_3 r^{\beta} \le C_2C’_3 r^{\sigma}
$$
Adding up, we deduce
$$
\phi (r) \le F(C_2,C_3,\beta,r,R) \le F(C_2,C’_3,\sigma,r,R)
$$
This justifies (2).
