I want to show that
$$\left( A + \frac{t}{x}N \right)^x \le \left( A + \frac{s}{y}N \right)^y $$
or equivalently
$$x\ln\left( A + \frac{t}{x}N \right) \le y\ln\left( A + \frac{s}{y}N \right)$$
We know that $x < y$ and $t+x = T$ where $T$ is a known constant. Also $s+y < T$, all the parameters and variables are non-negative.
I tried to optimize the function $f(x) = x\ln\left( A + \frac{T-x}{x}N \right)$ to show it's maximum value is always less than right hand side, but the solution to stationary points is not solvable, in other words
$$f'(x) = \ln\left( A + \frac{T-x}{x}N \right) - \frac{\frac{T}{x}N}{A + \frac{T-x}{x}N} = 0$$ has no closed form solution for $x$. How should I proceed in comparing these two? Is my approach correct? Is there another way? (BTW if this helps, $f(x)$ is convex).
Thanks in advance