I want to estimate $\frac{\tanh pa}{\tanh pb}$ like $\frac{\tanh pa}{\tanh pb}>\alpha$, where $p>0$ and $ a, b, \alpha$ is real number and $b>a$.
I have tried to find minimum of the function using derivative approach, but failed.
Another question is what if $p$ is complex number then how to handle the modulus of the function?
How to handle this type of function? any hint or help.
You can write your function as $$f(x) = \frac{\sinh(a+b)x -\sinh(b-a)x}{\sinh(a+b)x +\sinh(b-a)x}$$ This is an even function so has local extremum at x = 0, if Its limit at x = 0 exists. Furthermore, notice that $f$ is monotone increasing for positive x. Thus, its minima at x = 0 is global. Using the taylor aproximation $$\sinh(x) = x +o(x^3)$$ We get $$\lim_{x\rightarrow0} f(x) = \lim_{x\rightarrow0}\frac{(a+b)-(b-a)+o(x^3)/x}{(a+b)+(b-a) +o(x^3)/x} = a/b.$$
Thus, $f(x)<a/b$.
So the parity of the function, and its taylor series aproximation are useful sometimes to deal with such functions.