Abstract
Inthispaper, weintroducetheq-analogueofacertainfamilyoflinearoperatorsingeometricfunctiontheory. Our main purpose is to define some subclasses of analytic functions by means of the q-analogue of linear operators and investigate various inclusion relationships with integral preserving properties.
Highlights
Let A denote the class of analytic functions f (z) in the open unit disk O = {z : |z| < 1} such that ∞f (z) = z + anzn. n=2 (1.1)Subordination of two functions f and g is denoted by f ≺ g and defined as f (z) = g(w(z)), where w(z) is the Schwartz function in O
The q-difference operator, which was introduced by Jackson [12], is defined by f (z) − f
Motivated by the work mentioned above, we introduce the q-Srivastava–Attiya operator and the qmultiplier transformation on A as follows
Summary
The q-difference operator, which was introduced by Jackson [12], is defined by f (z) − f (qz). In [15], the authors introduced the q-analogue of the Ruscheweyh derivative operator and studied some of the properties of this differential operator. By using the concept of q-calculus, Arif et al [3] defined the q-Noor integral operator and investigated a number of important properties. Motivated by the work mentioned above, we introduce the q-Srivastava–Attiya operator and the qmultiplier transformation on A as follows. Using the q-derivative, we define some new classes of analytic functions below. Definition 1.1 A function f ∈ A is said to be in the class STq (φ) if it satisfies the following condition: zDqf (z) ≺ φ (z) , f (z) where Dq is the q-difference operator.
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