‎The main goal of this article is to present several quadratic refinements and reverses of the well known Heinz inequality‎, ‎for numbers and matrices‎, ‎where the refining term is a quadratic function in the mean parameters‎. ‎The proposed idea introduces a new approach to these inequalities‎, ‎where polynomial interpolation of the Heinz function plays a major role‎. ‎As a consequence‎, ‎we obtain a new proof of the celebrated Heron-Heinz inequality proved by Bhatia‎, ‎then we study an optimization problem to find the best possible refinement‎. ‎As applications‎, ‎we present matrix versions including unitarily invariant norms‎, ‎trace and determinant versions‎.

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