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