Exact estimates for integrals related to dirichlet series

First, we consider integrals of the form $$\int {a(x)m^{ - x} dx for} m = 2,3,...$$ over the unit interval (0, 1) or the interval (1, ∞) or the half-line (0, ∞), wherea(x)≥0 and is integrable on the interval in question. These integrals are related to the Dirichlet series $$\sum\limits_{m = 2}^\infty {a_m m^{ - x} for x > 1} ,$$ , where the numbersa _m ≥0. We survey certain known results in a new formulation in order to reveal the difference in behavior between the functions which are integrable on either (0, 1) or (1, ∞). Their proofs can be read out from the existing literature. Second, we extend these results from single to double related integrals, while making distinction among the functionsa(x, y) which are integrable on either (0, 1)² or (0, 1)×(1, ∞) or (1, ∞)×(0, 1) or (1, ∞)². The case wherea(x, y) is integrable on (0, ∞)² is also included.     

Statistical extension of classical Tauberian theorems in the case of logarithmic summability

Let s: [1,∞) → ? be a locally integrable function in Lebesgue’s sense. The logarithmic (also called harmonic) mean of the function s is defined by $$\tau (t): = \frac{1} {{\log t}}\int_1^t {\frac{{s(x)}} {x}dx, t > 1,}$$ where the logarithm is to the natural base e. Besides the ordinary limit lim _x→∞ s(x), we use the notion of the so-called statistical limit of s at ∞, in notation: st-lim _x→∞ s(x) = l, by which we mean that for every ? > 0, $$\mathop {\lim }\limits_{b \to \infty } \frac{1} {b}\left| {\left\{ {x \in (1,b):\left| {s(x) - \ell } \right| > \varepsilon } \right\}} \right| = 0.$$ We also use the ordinary limit limt→∞ τ (t) as well as the statistical limit st-limt→∞ τ (t). We will prove the following Tauberian theorem: Suppose that the real-valued function s is slowly decreasing or the complex-valued s is slowly oscillating. If the statistical limit st-limtt→∞ τ (t) = l exists, then the ordinary limit limx→∞ s (x) = l also exists.     

Investigation of the antibacterial mechanism of aflatoxins in the BioArena system

The influence of monomethylated basic amino acids [NG-monomethyl-L-arginine (MMA) and Nepsilon-monomethyl-L-lysine (MML)] and ozone capturers (indigo carmine, d-limonene) on the antibacterial effect of the mycotoxins aflatoxins B1, B2, G1 and G2 was studied in BioArena, which is a complex bioautographic system especially suitable for investigating biochemical interactions. In the presence of the formaldehyde precursors MMA or MML, the antibacterial-toxic activity of all the aflatoxins against the phytopathogenic bacterium Pseudomonas savastanoi pv. phaseolicola was enhanced dose-dependently. Indigo carmine and d-limonene, in appropriate concentrations, decreased the inhibition zones of aflatoxins. These results support the original idea that HCHO and its derivative 03 may be involved in the antibacterial activity of aflatoxins and so, potentially, in their known toxic effect.