Solutions of the equation aₙ + (a₍-₁ + (a₂ + (a₁ + x^r₁) ^r₂) ^r₍ = b\, x

We establish a novel upper bound for the real solutions of the equation specified in the title, employing a generalized derivation-division algorithm. As a consequence, we also derive a new set of Chebyshev functions adapted specifically for this problem.