Application of a subordination theorem associated with certain new generalized subclasses of analytic and univalent functions

The prime focus of the present work is to investigate some fascinating relations of some analytic and univalent functions using a subordination theorem.


Introduction
Let H denote the class of normalized analytic functions f (z) having the form: f (z) = z + a 2 z 2 + a 3 z 3 + ... (1) in the unit disk U = {z ∈ C : |z| < 1}. Also, let S denote the subclass of H univalent in U. Suppose that S * denote the subclass of S consisting of the functions f (z) which are starlike in U. A function f (z) ∈ K is said to be convex in U if f (z) ∈ S satisfies the condition that zf (z) ∈ S * . If f (z) ∈ H satisfies the geometric condition: for some real β(0 ≤ β < 1), then we say that f (z) belongs to the class S * (β) starlike of order β, and if f (z) ∈ H satisfies the geometric condition: for some real β(0 ≤ β < 1), then we say that f (z) belongs to the class K(β) convex of order β (see [1,2]). Let the function g(z) of the form: be in the class S * while the function g(z) of the form: (2020) 28:33 Page 2 of 10 be in the class K. With reference to (2) and (3), we can write that: where we consider the principal value of z kα for some real α (0 < α ≤ 2). See Darus and Owa [3] for some properties of functions f α (z) of the form (4).

Definition 1 (Subordination principle) For two functions f and g analytic in U, we say that f is subordinate to g, and write f ≺ g in U or f (z) ≺ g(z), if there exists a Schwarz function w(z), which is analytic in
Furthermore, if the function g is univalent in U: Also, we say that g(z) is superordinate to f (z) in U (see [4][5][6]).
Definition 2 (Subordinating factor sequence) A sequence {b k } ∞ k=1 of complex numbers is called subordinating factor sequence if for every univalent function f (z) in K, we have the subordination given by: [4][5][6]).
Lemma 2 Let s(z) (s(z) = 0) be a univalent function in U. Also, let μ = 0 be a complex number, then we have that: Suppose that r (r(z) = 0) satisfies the differential equation: then r ≺ s and s is the best dominant (see [8] among others).

Coefficient inequality
In this section, we consider the coefficient inequalities for function f α,n (z) given by (6) belonging to both classes S * α,n (A, B, γ ) and K α,n (A, B, γ ) in the unit disk U.

Theorem 2
Let the function f α,n (z) of the form (6) satisfy the inequality: The equality holds true for f α,n (z) given by: Proof The proof is similar to that of Theorem 1.

Remark 1 Putting A = n = 1 and B = −1 in Theorems 1 and 2, we obtain the results obtained by Darus and Owa [[3], Theorems 3 and 4].
Next, we present some subordination results.

Some subordination results
Our prime objective here is to establish sufficient conditions for functions belonging to the analytic class S * α,n (A, B, γ ).
Proof Suppose that we let: Then, Hamzat and El-Ashwah Journal of the Egyptian Mathematical Society (2020) 28:33 Page 5 of 10 and Using Lemma 2 in (18), then we obtain the desired result.
Proof Let f α,n ∈ S