On some properties of estimators of a probability density

M. A. Mirzakhmedov; Sh. A. Chashimov

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The connection between the functional inequalities $f (\frac{x + y}{2}) ⩽ \frac{f (x) + f (y)}{2} + α_{J} (x - y), x, y \in D,$ and $\int_{0}^{1} f (t x + (1 - t) y) ρ (t) d t ⩽ λ f (x) + (1 - λ) f (y) + α_{H} (x - y), x, y \in D,$ is investigated, where D is a convex subset of a linear space, f: D → ℝ, α H;α J: D-D → ℝ are even functions, λ ∈ [0; 1], and ρ: [0; 1] →ℝ+ is an integrable nonnegative function with ∫01 ρ(t) dt = 1.