On the Property of Monotonic Convergence for Beta Operators

Abstract

We show that beta operators satisfy the property of monotonic convergence under convexity. This gives a positive answer to a question recently posed by M. K. Khan. Some additional properties, consequences and applications are also discussed. Throughout this paper, probabilistic methods play a fundamental role.

On the Monotonicity of Positive Linear Operators

“…It was conjectured in [12] that (1) implies (2). This conjecture was recently proved by Adell et al [1]. And that (2) implies (3) is trivial.…”

“…For this result, the reader may see [1,3,4,12,14,16,18] and a list of further references in these papers. In particular, [14] gave a probabilistic method of showing that, under the condition of existence of first moment, K n have to be monotone from above for convex functions.…”

Strong Converse Results for Linking Operators and Convex Functions

“…For the proof, they used that B n,ρ can be written as a combination of the classical Bernstein operator and Beta operator and some corresponding results for the Beta operator from Adell et al [3], Theorem 1. For the case ρ = 1 and the case ρ = ∞, strong converse results are known [4], Theorem 1.1, [5], p.117 [6], and [7], Theorem 3.2, Theorem 5:…”

Best Constants in Preservation Inequalities Concerning the First Modulus and Lipschitz Classes for Bernstein-Type Operators

“…We consider the beta operator B t introduced by Mu hlbach [13] (see also [3,9]) for which we give the representation…”