Approximation operators and tauberian constants

The explicit expression of the smallest constant C satisfying {Mathematical expression} for all sequences {s _n} satisfying lim sup _n→∞ |d _n| <∞, where {t _n ⁽¹⁾ }, {t _n ⁽²⁾ } are two generalised Hausdorff transforms of {s _n }, {d _n} is the generalised (C, α)-transform (0≦α≦1) of {λ _n a _n} and n(λ, m(λ) are suitably related, is obtained. These results are obtained by using new properties of positive approximation operators and generalised Bernstein approximation operators.