I have function $$D(M\gamma)=\frac{M\gamma K}{K+\gamma(M\gamma-1)}\tag{1}$$ where $K$ is some positive constant. In this case, how to show that $$D(M(\gamma-1))<D(M\gamma),~~\text{for } \gamma<\bigg\lceil\sqrt{\frac{K}{M}}\bigg\rceil.\tag{2}$$
My Try:
I tried to show that $$D(M\gamma)-D(M(\gamma-1))>0$$ which after some manipulations results in following inequality $$\sqrt{\frac{K}{M}+1}>\gamma$$ But I think this does not mean that $$D(M(\gamma-1))<D(M\gamma),~~\text{for } \gamma<\bigg\lceil\sqrt{\frac{K}{M}}\bigg\rceil$$ because $$\sqrt{x+1}-\lceil\sqrt{x}\rceil$$ is not always greater than zero. So, How to prove the inequality in (2) for the function presented in (1). Any help in this regard will be much appreciated. Thanks in advance.