Approximation operators and tauberian constants

The explicit expression of the smallest constantC satisfying $$\mathop {lim}\limits_{\lambda \to \infty } \left| {t_{n(\lambda )}^{(1)} - t_{m(\lambda )}^{(2)} } \right| \leqq C. \mathop {lim sup}\limits_{n \to \infty } \left| {d_n } \right|$$ 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} andn(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.