Local and global results for modified Szász–Mirakjan operators

Abstract

In this paper, we study a natural modification of Szász–Mirakjan operators. It is shown by discussing many important established results for Szász–Mirakjan operators. The results do hold for this modification as well, be they local in nature or global, be they qualitative or quantitative. It is also shown that this generalization is meaningful by means of examples and graphical representations. Copyright © 2016 John Wiley & Sons, Ltd.

Harmonic starlikeness and convexity of integral operators generated by Poisson distribution series

“…Using the above series they obtain some interesting results on certain classes of analytic univalent functions. Some other interesting results also found in [5], [9] and [10], (see also [7], [11]). The convolution (or Hadamard product) of two series f (z) = ∞ n=0 a n z n and g(z) = ∞ n=0 b n z n is defined as the power series…”

“…, where H(z) and G(z) are given by (7). In order to show that I is locally univalent and sense-preserving it suffices to show that |H (z)| − |G (z)| > 0 in U .…”

Some approximation properties of a new class of linear operators

“…The generalized Jain operators as variant of the Lupaş operators were studied by Patel and Mishra . Some related work in this area can be found in other works . Motivated by this, we give further modification of Jain operators in this paper with the help of a generalized Poisson‐type distribution, establish its convergence properties, and discuss its degree of approximation, asymptotic formula, and statistical convergence.…”

“…15 Some related work in this area can be found in other works. [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] Motivated by this, we give further modification of Jain operators in this paper with the help of a generalized Poisson-type distribution, establish its convergence properties, and discuss its degree of approximation, asymptotic formula, and statistical convergence. The results for the Szász-Mirakyan operators and Jain operators can be obtained from our operators as a particular case.…”

A new approach to nonlinear singular integral operators depending on three parameters

“…For some studies concerning double singular integral operators in several settings, we refer the reader to [26][27][28][29][30]. On the other hand, for some other important works related to approximation by linear and nonlinear operators in several function spaces, we refer the reader to [31][32][33][34][35][36][37].…”