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We present a rational extension of Newton diagram for the positivity of${}_1F_2$ generalized hypergeometric functions. As an application, we giveupper and lower bounds for the transcendental roots $\beta(\alpha)$ of\begin{align*} \int_0^{j_{\alpha, 2}} t^{-\beta} J_\alpha(t) dt =0\qquad(-1<\alpha\le 1/2), \end{align*} where $j_{\alpha, 2}$ denotes thesecond positive zero of Bessel function $J_\alpha$.

Rational extension of Newton diagram for the positivity of ${}_1F_2$ hypergeometric functions and Askey-Szeg\"o problem