Operation with concentration inequalities and Conjugate of parallel sum

Following the concentration of the measure approach, we consider the transformation Φ(Z) of a random variable Z having a general concentration function α. If the transformation Φ is λ-Lipschitz with λ deterministic, the concentration function of Φ is immediately deduced to be equal to α(·/λ). If the variations of Φ are bounded by a random variable Λ having a concentration function (around $0$) β: R₊→ R, this paper sets that Φ(Z) has a concentration function analogous to γ = (α⁻¹ · β⁻¹)⁻¹. We apply this result to generalize Hanson Wright inequality, we express γ with the conjugate functions of the parallel sum of α and β and further consider the case where Λ is the product of n random variables to provide useful insights to the so-called ``multilevel concentration''.