Iosif Pinelis

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Elsewhere we developed rules for the monotonicity pattern of the ratio r := f/g of two differentiable functions on an interval (a, b) based on the monotonicity pattern of the ratio ρ := f ′/g′ of the derivatives. Those rules are applicable even more broadly than l’Hospital’s rules for limits, since in general we do not require that both f and g, or either(More)
Let (S0, S1, . . . ) be a supermartingale relative to a nondecreasing sequence of σ-algebras (H60,H61, . . . ), with S0 6 0 almost surely (a.s.) and differences Xi := Si − Si−1. Suppose that for every i = 1, 2, . . . there exist H6(i−1)-measurable r.v.’s Ci−1 and Di−1 and a positive real number si such that Ci−1 6 Xi 6 Di−1 and Di−1−Ci−1 6 2si a.s. Then for(More)
An algorithmic description of the dependence of the oscillation pattern of the ratio f g of two functions f and g on the oscillation pattern of the ratio f ′ g′ of their derivatives is given. This tool is then used in order to refine and extend the Yao-Iyer inequality, arising in bioequivalence studies. The convexity conjecture by Topsøe concerning(More)
Let BS1, . . . ,BSn be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter p ∈ (0, 1). Let m∗(p) := (1 + p+ 2 p2)/(2 √ p − p2 +4 p2) if 0 < p 6 1 2 and m∗(p) := 1 if 1 2 6 p < 1. Let m > m∗(p). Let f be such a function that f and f ′′ are nondecreasing and convex. Then it is proved that for(More)
Let (S0, S1, . . . ) be a supermartingale relative to a nondecreasing sequence of σ-algebras H≤0, H≤1, . . . , with S0 ≤ 0 almost surely (a.s.) and differences Xi := Si − Si−1. Suppose that Xi ≤ d and Var(Xi|H≤i−1) ≤ σ 2 i a.s. for every i = 1, 2, . . . , where d > 0 and σi > 0 are non-random constants. Let Tn := Z1 + · · · + Zn, where Z1, . . . , Zn are(More)
A convex polygon is defined as a sequence (V0, . . . , Vn−1) of points on a plane such that the union of the edges [V0, V1], . . . , [Vn−2, Vn−1], [Vn−1, V0] coincides with the boundary of the convex hull of the set of vertices {V0, . . . , Vn−1}. It is proved that all sub-polygons of any convex polygon with distinct vertices are convex. It is also proved(More)