On the convergence of a kind of q-gamma operators

Abstract

In this paper, we introduce a kind of q-gamma operators based on the concept of a q-integral. We estimate moments of these operators and establish direct and local approximation theorems of the operators. The estimates on the rate of convergence and weighted approximation of the operators are obtained, a Voronovskaya asymptotic formula is also presented. MSC: 41A10; 41A25; 41A36

P38 and JNK signal pathways are involved in the regulation of phlorizin against UVB‐induced skin damage

“…Therefore, ROS have been implicated in cutaneous ageing as well as in the pathogenesis of inflammatory skin diseases and skin cancer (8). Phlorizin has been reported to have antidiabetic effects due to its antioxidant properties (6). In our study, we observed that phlorizin can inhibit the overproduction of ROS induced by UVB irradiation in skin keratinocytes.…”

“…Phlorizin (phloretin-2-O-glucoside), a dihydrochalcone derived from apple bark, is known to produce renal glycosuria and block intestinal glucose absorption through inhibition of sodium/glucose cotransporters in the kidney and intestine (6). Recently, phlorizin and its analogues are being investigated for the treatment of diabetes, obesity and stress hyperglycaemia (7).…”

( p , q ) $(p,q)$ -gamma operators which preserve x 2 $x^{2}$

“…Later, many researchers have performed studies in this field, and the q-analogue of classical operators and modified operators, such as q-Szász-Mirakjan operators [28], q-Baskakov operators [13], q-Meyer-König-Zeller operators [12], q-Bleimann-Butzer-Hann operators [11] q-Phillips operators [29], q-Baskakov-Kantorovich operators [20], q-Baskakov-Durrmeyer operators [19], q-Szász-beta operators [18], and q-Meyer-König-Zeller-Durrmeyer operators [15], has been constructed; see also [2]. In [6], Cai and Zeng defined q-gamma operators…”

“…In addition, many well-known approximation operators with (p, q)-integer, such as (p, q)-Bernstein-Stancu-Schurer-Kantorovich operators [8], (p, q)-Szász-Baskakov operators [16], (p, q)-Baskakov-beta operators [30] have been introduced. All this achievement motivates us to construct the (p, q)-analogue of the gamma operator (1), as we know that many researchers have studied approximation properties of the gamma operators and their modifications (see [7,9,24,39]). The rest of the paper is organized as follows.…”

STANCU TYPE GENERALIZATION OF MODIFIED GAMMA OPERATORS BASED ON q-INTEGERS

“…in some function spaces. Very recently, Cai and Zeng [2] proposed a kind of modified q-generalization of Gamma operators and studied their approximation properties. The modified q-Gamma operators were defined in [2] as follows:…”