Abstract
This article deals with the q -differential subordinations for starlike functions associated with the lemniscate of Bernoulli and cardioid domain. The primary goal of this work is to find the conditions on γ for 1 + γ z ∂ q h z / h n z ≺ 1 + z , where h z is analytic function and is subordinated by the function which is producing cardioid domain as its image domain while mapping the open unit disk. Along with this, certain sufficient conditions for q -starlikeness of analytic functions are determined.
Highlights
Consider the class A of analytic functions defined in open unit disk F with normalization condition f ð0Þ = 0 and f ′ ð0Þ = 1 which provides the Taylor series expansion of the form ∞f ðzÞ = z + 〠 anzn, z ∈ F: ð1Þ n=2The class S consists of functions from A which are univalent functions in F, and the class P contains the analytic functions whose codomains are bounded by the open right half plane
The concept of differential subordination plays a vital role in the study of geometric properties of analytic functions
Many researchers contributed in the study of differential subordinations
Summary
[8] used the concept of differential subordination to prove analytic functions to be Janowski starlike. They proved differential subordinations with q-analogue for cardioid and limacon domain with the involvement of Janowski function and found the sufficient conditions for q-starlike functions. The q-derivative of a complex-valued function f , defined in the domain F, is given as follows: ÀÁ Dq f ðzÞ
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