Comparison of two Fourier transforms

I am looking for $delta>0$, such that

$$
delta int_{-infty}^{infty} exp(its)
{ Gamma{2(it+1)/3}over Gamma{(it+1)/2} }dt le \
int_{-infty}^{infty} exp(its)
{ Gamma (it+1)over Gamma{(it+1)/2} }dt
$$
for any $s>0$.

Ideally I want to find $delta(beta)$ such that
$$
delta(beta_1) int_{-infty}^{infty} exp(its)
{ Gamma{(it+1)/beta_1}over Gamma{(it+1)/2} }dt le \
delta(beta_2) int_{-infty}^{infty} exp(its)
{ Gamma {(it+1)/beta_2}over Gamma{(it+1)/2} }dt
$$
for any $s>0$, $1<beta_2<beta_1<2$. However it is far more complicated problem. Any hints will be greatly appreciated