How to prove $\frac{1}{m+n+1}-\frac{1}{(m+1)(n+1)} ≤ \frac{4}{45}$?

Question

If $m$ and $n$ are any two positive integers then how do we show that the inequality $$\frac{1}{m+n+1}-\frac{1}{(m+1)(n+1)} ≤ \frac{4}{45}$$ always holds ?

Answer 1

let $$f(m,n)=\dfrac{1}{m+n+1}-\dfrac{1}{(m+1)(n+1)}$$ then we easy to $$f(1,1)=f(1,2)=f(2,1)=\dfrac{1}{12}\le\dfrac{4}{45}$$ so,when $m,n\ge 2,$,then we let $j=m+n+2\ge 6$

then $$f(m,n)=\dfrac{1}{m+n+1}-\dfrac{1}{(m+1)(n+1)}\le\dfrac{1}{j-1}-\dfrac{4}{j^2}\le\dfrac{4}{45}$$ It is clearly, because $$f(x)=\dfrac{1}{x-1}-\dfrac{4}{x^2}\Longrightarrow f'(x)=-\dfrac{1}{(x-1)^2}+\dfrac{8}{x^3}=\dfrac{8(x-1)^2-x^3}{x^3(x-1)^2}<0,x\ge 6$$let where $j\ge 6$.