Summary: Given a probability measure \(\mu \) on Borel sigma-field of \(\mathbb R^d\), and a function \(f: \mathbb R^d \mapsto \mathbb R\), the main issue of this work is to establish inequalities of the type \(f(m)\leqslant M\), where \(m\) is a median (or a deepest point in the sense explained in the paper) of \(\mu \) and \(M\) is a median (or an appropriate quantile) of the measure \(\mu _f=\mu \circ f^{ - 1}\). For the most popular choice of halfspace depth, we prove that Jensen’s inequality holds for the class of quasi-convex and lower semi-continuous functions \(f\). To accomplish the task, we give a sequence of results regarding the “type \(D\) depth functions” according to the classification of Y. Zuo and R. Serfling [General notions of statistical depth function. Ann. Stat. 28, No. 2, 461–482 (2000; Zbl 1106.62334)], and prove several structural properties of medians, deepest points and depth functions. We introduce a notion of a median with respect to a partial order in \(\mathbb R^d\) and we present a version of Jensen’s inequality for such medians. Replacing the means in the classical Jensen inequality with medians gives rise to applications in the framework of Pitman’s estimation.