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Abstract The Poisson type inequalities, which were improved by Shu, Chen, and Vargas-De-Teón (J. Inequal. Appl. 2017:114, 2017), are generalized by using Poisson identities involving modified Poisson kernel functions with respect to a cone. New generalizations of improved Poisson–Sch type inequalities are obtained by using the generalized Montgomery identity associated with the Schrödinger operator. As applications in quantum calculus, we estimate the size of weighted Schrödingerean harmonic Bergman functions in the upper half space.