On q-Calculus and Starlike Functions

Abstract

We consider the class S Ã ðf; aÞ, 0 a\1, of normalized analytic functions f such that Re zd f f ðzÞ f ðzÞ & ' [ a; jzj\1; where d f f is the convolution operator d f f ðzÞ ¼ 1 z f ðzÞ Ã z ð1 À fzÞð1 À zÞ & ' ; where f is complex, jfj 1. For f ¼ 1 the operator becomes the derivative f 0 , while for real f ¼ q, 0\q\1, we obtain the Jackson q-derivative d q f .

q-Calculus and the Symmetrized Radioactive Decay Law

“…(1.8) is replaced by the number "1" and hence we get the well-known Jackson derivative in ref. [6][7][8][9] which is given by Eq. (1.9)…”

Properties for a Certain Subclass of Analytic Functions Associated with the Salagean q-Differential Operator

“…Based on the same idea, many authors have extensively studied the q-calculus operators (q-differential and q-integral operators) in GFT. A recent study on these operators acting on analytic functions can be found in [12][13][14][15][16][17][18][19]. For 0 < q < 1, Jackson [9,10] defined the q-differential operator, D q , of a function, ξ, as the following:…”

Applications of Some Subclasses of Meromorphic Functions Associated with the q-Derivatives of the q-Binomials

“…In recent years, the q-derivative operator D q was extensively used to create a number of analytic function classes (see, for examples [19][20][21][22][23][24][25]). In this paper, we construct inclusion relations for the families of meromorphic functions which we have introduced and described above.…”