New bounds for the function involving incomplete gamma function (Q2314636)

Let \[A_p(x)= \left(\int^x_0 e^{-t^p} dt\right)/x,\quad B_q(x)= 1-(1-e^{-qx^p})/(q(p+1)),\] with \(p,q>0\). The author considers among others bounds for \(A_p(x)- B_q(x)\) or \(A_p(x)/B_r(x)\), where \(r=1/(p+1)\). A main method applied in proofs is the L'Hôpital rule of monotonicity of functions.