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→+∞.