Applications of a Multiplier Transformation and Ruscheweyh Derivative for Obtaining New Strong Differential Subordinations

: Here, we study strong differential subordinations for the extended new operator IR m λ , l deﬁned by the Hadamard product of the extended multiplier transformation I ( m , λ , l ) and the extended Ruscheweyh derivative R m , on the class of normalized analytic functions A ∗ n ζ = { f ∈ H ( U × U ) , f ( z , ζ ) = z + a n + 1 ( ζ ) z n + 1 + . . . , z ∈ U , ζ ∈ U } , by IR m λ , l : A ∗ n ζ → A ∗ n ζ , IR m λ , l f


Introduction
Different types of operators have been used from early on in the study of complex functions. Among the advantages of using operators is the possibility of giving easier proofs of already known results but also facilitating the emergence of new, original research. The best-known operators are integral and differential operators. The first integral operator was introduced in 1915 [1] by Alexander. A very well-known integral operator was introduced by Libera in 1965; Bernardi then generalized it in 1969 [2]. Some of the best-known differential operators are the one used for obtaining the original results of the present paper, introduced by Ruscheweyh in 1975 [3], and Sȃlȃgean's differential operator, introduced in 1983 [4]. Recent studies have been conducted combining these two kinds of operators, obtaining differential-integral operators, such as seen in [5][6][7]. Convolution operators having numerous applications have been introduced, such as the Dziok-Srivastava operator [8], Srivastava-Wright operator [9], the operator introduced in [10] and the operator used in the present paper, defined in [11]. The monograph [12] and the paper [13], as well as the research performed in [14], give hints as to how the differential operator method is linked to partial differential equations and their applications.
The concept of differential subordination was introduced in [18,19] by S.S. Miller and P.T. Mocanu. The main results related to the theory of differential subordination can be found in [20] and we next recall some basic definitions as given in this monograph.
. The function f is said to be subordinate to F, if there exists a Schwarz function w, analytic in U, with w(0) = 0 and |w(z)| < 1, z ∈ U, such that f (z) = F(w(z)), z ∈ U. In such a case, we write f ≺ F. If F is univalent, then f ≺ F if and only if f (0) = g(0) and f (U) ⊂ g(U).

Definition 3. ([20])
Let ψ : C 3 × U → C and let h be univalent in U. If p is analytic and satisfies the differential subordination then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or, more simply, a dominant, if p ≺ q for all p satisfying (1). A dominant q that satisfies q ≺ q for all dominants q of (1) is said to be the best dominant of (1).
In studying the strong differential subordinations, we will use the following lemmas.

Lemma 1. ([21]
) Let h(z, ζ) be a convex function with h(0, ζ) = a for every ζ ∈ U and let γ ∈ C * be a complex number with Reγ ≥ 0. If p ∈ H * [a, n, ζ] and where α > 0 and n is a positive integer. If is holomorphic in U × U and and this result is sharp.
is defined by the following infinite series:
The author also extended in [11] the differential operator obtained as a convolution product (Hadamard product) of multiplier transformation and the Ruscheweyh derivative ( [25,26]) to the class A * nζ .
Definition 6. ( [11]) Let λ, l ≥ 0 and m ∈ N. Denote by IR m λ,l the extended operator given by the Hadamard product of the extended multiplier transformation I(m, λ, l) and the extended Ruscheweyh derivative R m , IR m λ,l : A * nζ → A * nζ , Remark 5. For l = 0, λ ≥ 0, we obtain the operator DR n λ studied in [27], and for l = 0 and λ = 1, we obtain the operator SR n studied in [28].
The symmetry properties of the functions used in defining an equation or inequality could be studied to determine solutions with particular properties. Regarding the differential subordinations or strong differential subordinations, which are some inequalities, the study of special functions, given their symmetry properties, could provide interesting results. Studies on the symmetry properties for different types of operators associated with the concept of quantum computing could also be investigated in a future paper.