Exploring the Validity of a Complex Inequality with Gamma Functions

[eljose]

let be the inequality:

$$[\frac{\Gamma(1/4-b/2-it/2)}{\Gamma(1/4+b/2+it/2)}]<\frac{\Gamma(1/4-b/2)}{\Gamma(1/4+b/2)}$$

where [] means modulus of the complex number...

 

[matt grime]

and gamma is the gamma function presumably... is b any real or complex number? and t? apparently b must be real, looking at it. i mean, is the rhs modded too?

 

[eljose]

b and t are both real and $$\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}dt$$

 

[shmoe]

It's certainly not true for any b, remember the asymptotic I gave you for $$\chi$$ before? It can easily tell you which b's even have a chance for this to be true and also that for these b's it will hold for a large enough |t| (though not uniformly).