Abstract
In this article, we study differential subordnations in q-analogue. Some properties of analytic functions in q-analogue associated with cardioid domain and limacon domain are considered. In particular, we determine conditions on α such that 1 + α z ∂ q h z h z n ( n = 0 , 1 , 2 , 3 ) are subordinated by Janowski functions and h z ≺ 1 + 4 3 z + 2 3 z 2 . We also consider the same implications such that h z ≺ 1 + 2 z + 1 2 z 2 . We apply these results on analytic functions to find sufficient conditions for q-starlikeness related with cardioid and limacon.
Highlights
We recall here some basic notions from the literature of Geometric Function Theory which are essential for clarity and understandings of the upcoming work
We start with the symbol A which represents the family of analytic functions in D = {z : |z| < 1} and any function f in A satisfies the conditions f (0) = 0 and f 0 (0) − 1 = 0
The notion of subordinations between analytic functions is represented by f ≺ g and is defined as; a function f is subordinated by function g, if we can find an analytic function w with the properties w (0) = 0 and
Summary
We recall here some basic notions from the literature of Geometric Function Theory which are essential for clarity and understandings of the upcoming work. By choosing some more particular function φ(z), we get several interesting subclasses of starlike functions. Using the definition of q-derivative, Seoudy and Aouf [29] introduced the class Sq∗ ( φ) For this class, the familiar Fekete-Szegö problem was obtained by the authors. By choosing particular functions instead of the function φ, we obtain several interesting subclasses of starlike functions associated with different image domains. Similar kind of differential subordinations are discussed by various authors They used these results to find sufficient conditions for starlike functions, see [31,32,33,34,35,36]. Similar results are obtained for h (z) ≺ 1 + 2z + 12 z2 , z ∈ D We use these results to find sufficient conditions for q-starlike functions associated with cardioid and limacon.
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