Perturbation-theoretic justification of Akra–Bazzi bounds for shifted divide-and-conquer recurrences

Develop a general perturbation theorem that, without invoking the Akra–Bazzi result, establishes explicit sufficient conditions on the shift functions h(x,t) under which every solution of the perturbed divide-and-conquer recurrence T(x) = ∫_{m}^{M} T(x/t + h(x,t)) dμ(t) + g(x), with μ a finite positive measure supported on [m, M] for 1 < m ≤ M < ∞, g a nonnegative O-regularly varying function, and p the unique nonnegative solution of ∫_{m}^{M} t^{-p} dμ(t) = 1, satisfies the Akra–Bazzi asymptotic bound T(x) = Θ(x^p (1 + ∫_{x0}^{x} t^{-p} g(t) dt/t)).

Background

The paper studies general forms of the Master Theorem for divide-and-conquer recurrences and presents a streamlined proof of the Akra–Bazzi bound, extending it to handle floors and ceilings via additive shifts encoded by functions h(x,t).

In discussing the recursion T(x) = ∫ T(x/t + h(x,t)) dμ(t) + g(x), the author notes that this looks like a perturbation of the unshifted case h ≡ 0 and suggests that a general perturbation principle should imply the same asymptotic behavior as Akra–Bazzi under suitable constraints on h(x,t). However, no such general theorem is currently known, prompting the author to proceed via a direct inductive proof instead.

The open problem is to formulate and prove a general perturbation-theoretic result that directly yields the Akra–Bazzi asymptotics for the perturbed recursion under explicit, natural conditions on h(x,t), thereby providing a conceptually simpler route than the detailed inductive arguments.