This paper proves a generalization of M. Riesz's convexity theorem using interpolation theory for linear operators. The main result is Theorem 1, which provides an interpolation theorem for analytic families of operators with admissible growth. The theorem states that if certain bounds on the operator norms at the endpoints of the strip are satisfied, then the operator norm at any intermediate point can be bounded by a convex combination of the endpoint bounds. The proof uses Hirschman's lemma and the three-lines lemma. The paper also presents two applications of the interpolation theorem. In Part II, it proves mean convergence of Bochner-Riesz means for multiple Fourier integrals and series below the critical index. In Part III, it generalizes Pitt's theorem for orthonormal systems, showing that it holds for all uniformly bounded orthonormal systems, not just Fourier series. The result includes known inequalities of Riesz, Paley, and Hirschman. The proof of Pitt's theorem uses interpolation techniques and Paley's inequalities. The paper concludes with references to related works by various authors.This paper proves a generalization of M. Riesz's convexity theorem using interpolation theory for linear operators. The main result is Theorem 1, which provides an interpolation theorem for analytic families of operators with admissible growth. The theorem states that if certain bounds on the operator norms at the endpoints of the strip are satisfied, then the operator norm at any intermediate point can be bounded by a convex combination of the endpoint bounds. The proof uses Hirschman's lemma and the three-lines lemma. The paper also presents two applications of the interpolation theorem. In Part II, it proves mean convergence of Bochner-Riesz means for multiple Fourier integrals and series below the critical index. In Part III, it generalizes Pitt's theorem for orthonormal systems, showing that it holds for all uniformly bounded orthonormal systems, not just Fourier series. The result includes known inequalities of Riesz, Paley, and Hirschman. The proof of Pitt's theorem uses interpolation techniques and Paley's inequalities. The paper concludes with references to related works by various authors.