Stability of Some Positive Linear Operators on Compact Disk

TL;DR

This paper investigates the Hyers-Ulam stability of various complex positive linear operators on the compact disk, extending previous stability results from [0,1] to more complex operators.

Contribution

It establishes Hyers-Ulam stability for complex Bernstein-Schurer, Kantrovich-Schurer, and Lorentz operators on the compact disk, and determines the minimal stability constants.

Findings

Proves Hyers-Ulam stability for the specified operators.

Determines the infimum of stability constants for stable operators.

Extends stability analysis from [0,1] to complex operators on the disk.

Abstract

Recently, Popa and Rasa [18,19] have shown the (in)stability of some classical operators defined on [0,1] and found best constant when the positive linear operators are stable in the sense of Hyers-Ulam. In this paper we show Hyers-Ulam (in)stability of complex Bernstein-Schurer operators, complex Kantrovich-Schurer operators and Lorentz operators on compact disk. In the case when the operator is stable in the sense of Hyers and Ulam, we find the infimum of Hyers-Ulam stability constants for respective operators.