# Proving complicated transcendental inequality

Suppose we have a function $$f$$ of four posirive real numbers $$a,b,c$$ and $$d$$ in a domain that, for a given real number $$0 they satisfy

$$rc $$rc

We then have

$$f(a,b,c,d)=\frac{\log\sinh d-\log\sinh c+c\coth b-d\coth d}{\log\sinh a-\log\sinh b+b\coth b-a\coth d}.$$

I have been tasked to show that, in the given domain, $$f(a,b,c,d)\geq1.$$

To check that it was even reasonable, I put it on Mathematica and tried it for a few values of $$r$$ and indeed the minimum value was 1 (I had to work with 1000 digit precision, however). With this in mind, I then tried taking the gradient of the function and setting it equal to zero. This then led to a transcendental equation, which I haven't been able to solve and I'm not even sure that the solutions would be in the domain.

What other method could there be to prove this inequality?