# Trigonometric inequality for Theorem 3.4.4 in Stenger 1993

I'm looking at the proof of Theorem 3.4.4 in "Frank Stenger Numerical Methods based on Sinc and Analytic Functions". The book seems to be available on google books, but I will try to make the specific issue self-contained.

Let $$t,d,h \in \mathbb{R}$$ and $$z\in \mathbb{C}$$, write $$z=x + iy$$. Assume $$-d. Then I need the bound $$\left \lvert \frac{ e^{-i \pi (t-id)/h} - \cos(\pi z /h)}{ (t-id-z) \sin( \pi (t-id)/h)}-\frac{ e^{-i \pi (t+id)/h} - \cos(\pi z /h)}{ (t+id-z) \sin( \pi (t+id)/h)} \right \lvert \leq \frac{ e^{-\pi d/h} + \cosh(\pi y/h)}{ d \sinh(\pi d / h)}$$

It is clear that $$\lvert e^{-i \pi (t-id)/h} \lvert \leq e^{- \pi d /h}$$ and that $$\lvert \cos(\pi z /h)) \lvert \leq \cosh(\pi y/h)$$, so the triangle inequality gets us some way. I can't seem to get anywhere with the denominator though.

First $|t\pm id-z|\ge |d|$. Also for real $a$ and $b$ $$|\sin(a+ib)|=|\sin a \cosh b+i \cos a \sinh b|=|\sin a\, \text{coth}\, b+i \cos a| |\sinh b |\ge |\sinh b|$$ since $|\text{coth}\, b|\ge 1$. This leads to the rhs $$2\frac{ e^{-\pi d/h} + \cosh(\pi y/h)}{ |d \sinh(\pi d / h)|}.$$