Quasi-Hadamard Product of Certain ω-Starlike and ω-Convex Functions with respect to Symmetric Points

The main purpose of this paper is to derive some results associated with the quasi-Hadamard product of certain -starlike and -convex univalent analytic functions with respect to symmetric points.

Let  *  be the subclass of () consisting of functions given by (1) with  = 0 and satisfying the condition These functions are called starlike with respect to symmetric points and were introduced by Sakaguchi [5].Motivated by the previous classes  *  , Oladipo [6] (see also [7]) defined the following classes of functions with respect to symmetric points.
and  is a fixed point in .These functions are called starlike with respect to symmetric points.
(ii) Let    () be the subclass of () consisting of functions given by (1) satisfying the condition Re ( (( − )   ()) and  is a fixed point in .These functions are called convex with respect to symmetric points.Suppose that  and  are two analytic functions in .Then, we say that the function  is subordinate to the function , and we write if there exists a Schwarz function () with (0) = 0 and |()| < 1 such that By applying the previous subordination definition, we define the following subclasses of  *  () and    ().
(ii) Let    (, , ) be the subclass of () consisting of functions given by (1) satisfying the condition where −1 ≤  <  ≤ 1,  ∈ , and  is a fixed point in .

Lemma 3. A function 𝑓 defined by (2) belongs to the class
Now, we introduce the following class of analytic functions in .
Definition 5. A function  of the form (2), which is analytic in , belongs to the class  * , (, , ), if it satisfies the condition where −1 ≤  <  ≤ 1 and  is any fixed nonnegative real number.
We note that, for any nonnegative real number , the class  * , (, , ) is nonempty as the functions of the form where  1 > 0,   ≥ 0, and Similarly, we can define the quasi-Hadamard product of more than two functions; for example, where the functions   ( = 1, 2, . . ., ) are given by (3).

Main Results
Unless otherwise mentioned, we will assume throughout the following results that −1 ≤  <  ≤ 1,  ∈ ,  is any fixed nonnegative real number, and  is a fixed point in .
By taking  = 0 in Theorem 8, we get the following result.Next, we discuss some applications of Theorems 6 and 8.