A Generalization of the Wang–Ahmad Inequality

Asymptotically exact constants in natural convergence rate estimates in the Lindeberg theorem

“…The main goal of the present work is construction of the lower bounds of the absolute constants C E (ε, γ), C R (ε, γ) in inequalities (11), (12), and also of the constants A E (ε, γ), A R (ε, γ) in inequalities (1), (2), in particular, we show that even in the i.i.d. case…”

“…The constants C E (g, ε, γ) and C R (g, ε, γ) are the minimal possible (exact) values of the constants C E (ε, γ) and C R (ε, γ) in inequalities (11), (12) for the fixed function g ∈ G, while their universal values sup g∈G C E (g, ε, γ), sup g∈G C R (g, ε, γ), that provide the validity of the inequalities under consideration for all g ∈ G are called exact constants and namely they are the minimal possible (exact) values of the constants C E (ε, γ) and C R (ε, γ) in ( 11) and (12), respectively. In order not to introduce excess notation and following the above convention, we use namely these exact values for the definitions of C E (ε, γ) and C R (ε, γ) in the present work:…”

“…Let us denote by G a set of all non-decreasing functions g : [0, ∞) → [0, ∞) such that g(z) > 0 for z > 0 and z/g(z) is also non-decreasing for z > 0. The set G was initially introduced by Katz [15] and used later in the works [20,18,17,11,10,12]. In [11] it was proved that (i) For every function g ∈ G and a > 0…”

“…The set G was initially introduced by Katz [15] and used later in the works [20,18,17,11,10,12]. In [11] it was proved that (i) For every function g ∈ G and a > 0…”

“…Inequality (11) also generalizes and improves up to the values of the appearing absolute constant the classical Katz-Petrov inequality [15,20] (which is equivalent to the Osipov inequality [19]) because of involving the algebraic truncated third order moments instead of the absolutes ones and also a recent result of Wang and Ahmad [26] at the expense of moving the modulus and the least upper bound signs outside of the sum sign. In particular, inequality (11) established an upper bound of the constant in the Wang-Ahmad inequality C E (∞, 1) A E (1, 1) 2.73.…”