I'm trying to prove the following inequality:
For $x \in (0,3)$, $$ {_1F_2[1;\frac{5}{4},\frac{7}{4};\frac{-(1\cdot x)^2}{4}]}+{_1F_2[1;\frac{5}{4},\frac{7}{4};\frac{-(3\cdot x)^2}{4}]}\gt 2\cdot {_1F_2[1;\frac{5}{4},\frac{7}{4};\frac{-(5\cdot x)^2}{4}]}$$
My attempts:
I wrote Fresnel Integral Transform for Hypergeometric Functions, but it gaves me only more complicated formula to prove.
I also find some Series representations for Hypergeometric Functions. But I then tried unsuccessfully to express it and didn't find a good one to re-express it.
I also searched L.Luke's book trying to use asymptote to show it but I didn't find a good approximation. Perhaps the hypergeom can simplify with the Bessel function?
Any help would be appreciated! Thanks!
This can be brute-forced by using the same approach as here.
The left-hand side of the inequality will have to be expanded to order 16, and the right-hand side to order 34; the approximation errors will be bounded by the absolute values of the $x^{18}$ and the $x^{36}$ terms respectively.
Then, since everything is polynomial, a Sturm sequence can be computed to prove that the difference of the lower bound for the left-hand side and the upper bound for the right-hand side does not have zeros on $(0,3]$.