Asymptotic form of the balanced model metric solution at infinity

Prove that for each parameter β ∈ [0,1), if f(x) is the unique analytic solution with f(0) = 1 to the functional equation ∫_0^∞ (f(sx)/f(x)) dx = 1/(1−s) − β for all s ∈ [0,1), then as x → ∞ one has the asymptotic f(x) ∼ C_β x^β e^x for some constant C_β depending on β.

Background

The paper continues the paper of logarithmic balanced model metrics initiated in earlier work by the authors. In the special case X = ℂ with divisor D at the origin, the balanced metric corresponds to an analytic function f solving a functional equation that encodes the balanced embedding condition.

Understanding the asymptotics of f(x) as x → ∞ is central to connecting the infinite-dimensional balanced embedding problem with compact settings. The authors explicitly state a conjecture that f(x) should have the asymptotic form C_β x^β e^x, and the present paper partially confirms a weaker form by establishing the existence of the exponent limit ω(β).