Logarithms of Laplace–Stiltjes integral and maximum of the integrand

Abstract

Let f(x)≥0, , μ(σ)=sup {f(x)e ^xσ: x≥0}, σ∈ℝ, and let Φ be a positive function on (−∞,+∞) such that Φ′(σ)↑+∞ as σ→+∞. We investigate the conditions on F, under which the asymptotical inequality ln μ(σ)≤(1+o(1))Φ(σ) implies the asymptotical inequality ln I(σ)≤(1+o(1))Φ(σ) as σ→+∞.

Keywords:

• Laplace–Stiltjes integral
• Maximum of the integrand