Bounds on the Gell-Mann--Low functions in quantum electrodynamics and in the Wess-Zumino model

We derive bounds $\mathrm{\mid }\frac{d\psi (\alpha )}{d\alpha }\mathrm{\mid }\le 1$, $\frac{d(\frac{d\psi (\alpha )}{d\alpha }\psi (\alpha ))}{d\alpha }\le 1$ on the GL (Gell-Mann–Low) function $\psi (\alpha )$ from the Kallen-Lehmann dispersion representation in quantum electrodynamics. We also derive analogous bounds for the GL function in the Wess-Zumino model. The implications of the obtained inequalities are discussed. In particular, we obtain bounds on coupling constants in dark photon model and in dark matter model with vector $(B-L)$ messenger.