On the Containment of the Unit Disc Image by Analytical Functions in the Lemniscate and Nephroid Domains

Suppose that A 1 is a class of analytic functions f : D = { z ∈ C : | z | < 1 } → C with normalization f ( 0 ) = 1 . Consider two functions P l ( z ) = 1 + z and Φ N e ( z ) = 1 + z − z 3 / 3 , which map the boundary of D to a cusp of lemniscate and to a twi-cusped kidney-shaped nephroid curve in the right half plane, respectively. In this article, we aim to construct functions f ∈ A 0 for which (i) f ( D ) ⊂ P l ( D ) ∩ Φ N e ( D ) (ii) f ( D ) ⊂ P l ( D ) , but f ( D ) ⊄ Φ N e ( D ) (iii) f ( D ) ⊂ Φ N e ( D ) , but f ( D ) ⊄ P l ( D ) . We validate the results graphically and analytically. To prove the results analytically, we use the concept of subordination. In this process, we establish the connection lemniscate (and nephroid) domain and functions, including g α ( z ) : = 1 + α z 2 , | α | ≤ 1 , the polynomial g α , β ( z ) : = 1 + α z + β z 3 , α , β ∈ R , as well as Lerch’s transcendent function, Incomplete gamma function, Bessel and Modified Bessel functions, and confluent and generalized hypergeometric functions.