In this short note our aim is to establish some Grünbaum-type inequalities for the complementary error function, the incomplete gamma function and for Mills’ ratio of the standard normal distribution, and of the gamma distribution, respectively.

We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it derive the of Incomplete gamma function terms Hurwitz zeta function. then evaluate this formula new series special functions fundamental constants. All results work are new.