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 $0<r<1$ they satisfy

$$rc<b<a,$$ $$rc<rd<a.$$

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?


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