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How to prove that $ sum_{m=0}^{infty} { Gamma{(1+2m)/alpha}over Gamma(1/2+m)} { (-t^2/4)^{m}over m !} ge (alpha/2)^{3}exp(-t^{2}/4) $

I would love to prove the following inequality
$$
{1over sqrt{pi} } sum_{m=0}^{infty}
Gamma{(1+2m)/alpha}
{ (-t^2)^{m}over (2m) !}=$$

$$
sum_{m=0}^{infty}
{ Gamma{(1+2m)/alpha}over Gamma(1/2+m)}
{ (-t^2/4)^{m}over m !} ge (alpha/2)^{3}exp(-t^{2}/4)
$$

$1<alpha<2$, $t>0$,
The question is connected to the other question I asked and got no answer for Prove $int_{0}^{infty} cos(omega x) exp(-x^{alpha}) , {rm d} x ge {alpha^2 sqrt{pi} over 8} exp left( -frac{omega^2}{4} right)$

MathOverflow Asked on January 3, 2022

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