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I want to know if the function below is monotonically decreasing for all $a,b >0, a\neq b $

\begin{equation} x\rightarrow \frac{\sinh^2((a-b)x)}{\sinh(2ax)\sinh(2bx)} \text{, $x >0. $} \end{equation}

Thank you for you response.

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  • $\begingroup$ What is squared in the numerator: ${\rm sinh}$ or its argument only? What is the motivation behind this question? (Sounds like a homework problem to me, and asking homework questions is strongly discouraged at this site.) $\endgroup$
    – Seva
    Commented Jan 9, 2016 at 18:59
  • $\begingroup$ @seva Hi , it is the whole $(\sinh((a-b)x))^2$. Actually it is not a homework problem. It's a green kernel from some operator that I want to study. I tried to study the derivative, but it is not obvious at all. $\endgroup$
    – MathGuy1991
    Commented Jan 9, 2016 at 19:09
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    $\begingroup$ It IS monotonically decreasing for $a=b,$ so no need to set $a\neq b.$ $\endgroup$
    – Igor Rivin
    Commented Jan 9, 2016 at 20:46
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    $\begingroup$ Take logarithms, take derivative, simplify using mathematica, it seems to work out. $\endgroup$
    – Suvrit
    Commented Jan 9, 2016 at 20:48
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    $\begingroup$ @Suvrit Why don't you take the derivative directly? $(\log f)'=f'/f$ has the same sign as $f'$. $\endgroup$
    – Fan Zheng
    Commented Jan 9, 2016 at 21:19

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