Majorization for a Class of Analytic Functions Defined by q-Differentiation

K. A. Selvakumaran, Sunil Dutt Purohit, and Aydin Secer 1 Department of Mathematics, R.M.K College of Engineering and Technology, Puduvoyal, Tamil Nadu 601206, India 2Department of Basic Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur, Rajasthan 313001, India 3 Department of Mathematical Engineering, Yildiz Technical University, Davutpasa, 34210 Istanbul, Turkey

For the convenience of the reader, we now give some basic definitions and related details of -calculus which are used in the sequel.
and as a special case of the above series for  = , we have Also, the -derivative and -integral of a function on a subset of C are, respectively, given by (see [2, pp. 19-22]) Therefore, the -derivative of () =   , where  is a positive integer, is given by where and is called the -analogue of .As  → 1, we have Here we list some relations satisfied by []  : Recently, many authors have introduced new classes of analytic functions using -calculus operators.For some recent investigations on the classes of analytic functions defined by using -calculus operators and related topics, we refer the reader to [3][4][5][6][7][8][9][10][11][12][13] and the references cited therein.In the following, we define the fractional -calculus operators of a complex-valued function (), which were recently studied by Purohit and Raina [9].
Definition 1 (fractional -integral operator).The fractional integral operator   , of a function () of order  is defined by where () is analytic in a simply connected region of the plane containing the origin and the -binomial function ( − ) −1 is given by The Definition 2 (fractional -derivative operator).The fractional -derivative operator   , of a function () of order  is defined by where () is suitably constrained and the multiplicity of ( − ) − is removed as in Definition 1.
Definition 3 (extended fractional -derivative operator).Under the hypotheses of Definition 2, the fractional derivative for a function () of order  is defined by where  − 1 ≤  < 1,  ∈ N 0 = N ∪ {0}, and N denotes the set of natural numbers.

Majorization Problem for the Class 𝑆 𝛿,𝑗 𝑞,𝑝 (𝑏)
We start by proving the following -analogue of the result given by Nehari in [17].