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Abstract We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including an extension of the Busemann–Petty problem and a slicing inequality for arbitrary functions. Let $$K$$ and $$L$$ be star bodies in $${\mathbb R}^n,$$ let $0<k<n$ be an integer, and let $f,g$ be non-negative continuous functions on $$K$$ and $$L$$, respectively, so that $$\|g\|_\infty =g(0)=1.$$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{n-k}n}|K|^{\frac kn}} \le \frac n{n-k} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $|K|$ stands for volume of proper dimension, $$C$$ is an absolute constant, the maximum is taken over all $(n-k)$-dimensional subspaces of $${\mathbb R}^n,$$ and $$d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)$$ is the outer volume ratio distance from $$K$$ to the class of generalized $$k$$-intersection bodies in $${\mathbb R}^n.$$ Another consequence of this result is a mean value inequality for the Radon transform. We also obtain a generalization of the isomorphic version of the Shephard problem.