Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that 
$$
\frac{y\left( x \right)}{p\left( x \right)}=\frac{\left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta}}{\mathbb{E} _{x\sim p}\left[ \left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta} \right]},
$$
or equivalently
$$
y\left( x \right) =\frac{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}{\sum_x{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}}
$$
Under these conditions, can this equality
$$
\mathrm{KL}\left( y||q \right) \ge \beta \mathrm{KL}\left( p||q \right) 
$$
holds? If not, what's the counter example?