Let $X$ be a random variable with pdf $f$. Suppose we want to test $$H_0:f=f_0\quad $$ against $$ H_1:f=f_1$$. For any $\alpha>0$, define
$$T_\alpha(X)= 1\qquad if\quad f_1(X)>c(\alpha)f_0(X)$$ $$\qquad=\gamma(\alpha)\qquad if\quad f_1(X)=c(\alpha)f_0(X)$$ $$\qquad=0\qquad if\quad f_1(X)<c(\alpha)f_0(X)$$
where $0\le\alpha\le1$, $c(\alpha)\geq 0$ and $E[T_\alpha(X)]=\alpha$ under $H_0$. Can we show
(i) if $\alpha_1\lt\alpha_2$ then $c(\alpha_1)\geq(\alpha_2)$
(ii) if $\alpha_1\lt\alpha_2$ then type II error probability of $T_{\alpha_1}(X)$ is larger than that of $T_{\alpha_2}(X)$.
When using $\alpha_1\le\alpha_2 \implies E[T_{\alpha_1}(X)]\le E[T_{\alpha_2}(X)]$ I am arriving at conclusions that are not at all informative. Please help.