Based on our previous modified log-wake law in turbulent pipe flows, we invent two compound similarity numbers (Y, U), where Y is a combination of the inner variable y+ and outer variable ξ, and U is the pure effect of the wall. The two similarity numbers can well collapse mean velocity profile data with different moderate and large Reynolds numbers into a single universal profile. We then propose an arctangent law for the buffer layer and a general log law for the outer region in terms of (Y, U). From Milikan's maximum velocity law and the Princeton superpipe data, we derive the von Kármán constant κ= 0.43 and the additive constant B ≈ 6. Using an asymptotic matching method, we obtain a self-similarity law that describes the mean velocity profile from the wall to axis; and embeds the linear law in the viscous sublayer, the quartic law in the bursting sublayer, the classic log law in the overlap, the sine-square wake law in the wake layer, and the parabolic law near the pipe axis. The proposed arctangent law, the general log law and the self-similarity law have been confirmed with the high-quality data sets, with different Reynolds numbers, including those from the Princeton superpipe, Loulou et al., Durst et al., Perry et al., and den Toonder and Nieuwstadt. Finally, as an application of the proposed laws, we improve the McKeon et al. method for Pitot probe displacement correction, which can be used to correct the widely used Zagarola and Smits data set.