Beta-type operators preserve shape properties

On the Property of Monotonic Convergence for Beta Operators

“…This probabilistic representation is quite different from (3) and was introduced in [1] to study preservation of monotonicity, convexity and Lipschitz constants. Since, for 0<x<1 and 0<r tx, the random variables U rx ÂU r and U tx ÂU t are independent (cf.…”

“…Notwithstanding, in view of (17) and taking into account that B t preserves monotonicity (cf. [1]), we deduce from Theorem 1:…”

“…This operator has been considered by several authors (see, for instance, [1,8,12,15]). A slight modification of B t is the operator B t * given by B(tx+1, t(1&x)+1) d%, t 0,…”

Note on a Class of Operators on Infinite Interval

“…Adell, F.G. Badia, Jesüs de la Call and M. San Miguel (see [1], [2], [3]). If / is defined in χ = 0 as well, we set (T Q /)(0) = /(0).…”

“…Lupa § and M. Müller introduced [5] an integral operator defined by (1) (Gn/)(x) = J gn(x, u)f du, n = 1,2,..., where…”