A New Parametric Differential Operator of p-Valently Analytic Functions

Newly, numerous investigations are considered utilizing the idea of parametric operators (integral and differential). The objective of this effort is to formulate a new 2D-parameter differential operator (PDO) of a class of multivalent functions in the open unit disk. Consequently, we formulate the suggested operator in some interesting classes of analytic functions to study its geometric properties. The recognized class contains some recent works.


Introduction
In analysis, a parametric differential operator (PDO) is a differential operator of a dependent variable with respect to another dependent variable that is engaged when both variables formulate on an independent third variable, typically supposed as "time." We shall use this idea to consider the PDO of a complex variable to discuss its properties in the opinion of the geometric function theory (GFT). The field of differential operators is investigated in GFT early by the well-known Salagean differential operator and the Ruscheweyh derivative. Later, these operators are generalized by different types of parameters using a 1D-parameter fractional differential operator [1] and 2D-parameter fractional differential operator [2]. Recently, using the class of normalized functions ψ ∈ Σ ψ ζ ð Þ = ζ + 〠 ∞ n=2 ψ n ζ n , ζ ∈ Δ ≔ ζ ∈ ℂ : ζ j j < 1 f g : Ibrahim and Jay [3] presented PDO of the following form: for α ∈ ½0, 1 P 0 ψ ζ ð Þ = ψ ζ ð Þ, The functions ρ 1 , ρ 0 : ½0, 1 × Δ ⟶ Δ are analytic in Δ satisfying ρ 1 ðα, ζÞ ≠ −ρ 0 ðα, ζÞ. More studies are given by Ibrahim and Baleanu [4,5] using (2) to present a hybrid diff-integral operator and a quantum hybrid operator, respectively.
In this effort, we generalize (2) by considering another class of analytic functions denoting by Σ ℘ and constructing by which are analytic in Δ: Recently, different investigations are presented studying the geometric behavior of this class (see [6][7][8][9]). The Hadamard product [10,11] for two functions in Σ ℘ is given by the series Definition 1. For a function ψ ∈ Σ ℘ , PDO is defined as follows: where ρ 1 and ρ 0 are defined in (3) and (4), respectively.

Remark 2.
(i) It is clear that Q m α ψðζÞ ∈ Σ ℘ , and it is a generalization of (2) (℘ = 1) (ii) The integral operator that corresponds to Q m α ψðζÞ is where Moreover, we have the following property: Proposition 3 (semigroup property). Consider the PDO; then for ψ and φ ∈ Σ ℘ Proof. Let m = 1; the definition of Q mν implies Hence, for all m, we have the desired assertion.
Our study is about the following class: where the symbol ≺ presents the subordination symbol [12] and p is convex univalent in Δ.

For example
which is univalent convex in Δ, and it is the extreme function in the set

Journal of Function Spaces
Define a functional Ψ : Δ ⟶ Δ, as follows: where Shortly, by Definition 4 we say Our aim is to study the operator formula Ψ. We recall the following results: Lemma 5 (see [12]). Let two analytic functions f ðζÞ and gðζÞ be convex univalent defined in Δ such that f ð0Þ = gð0Þ: Lemma 6 (see [12]). Define the general class of holomorphic functions where a ∈ ℂ and n is a positive integer. If c ∈ ℝ, then Moreover, if c > 0 and h ∈ ℍ½1, n, then there are fixed numbers ℓ 1 > 0 and ℓ 2 > 0 with the inequality Lemma 7 (see [13]). Let ℏ, p ∈ ℍ½a, n , where p is convex univalent in Δ and for k 1 , k 2 ∈ ℂ, k 2 ≠ 0 ; then Lemma 8 (see [14]). Let h, p ∈ ℍ½a, n , where p is convex univalent in Δ such that hðζÞ + kζh ′ ðζÞ is univalent; then Lemma 9 (see [15]). Let ℏ, y, g ∈ ℍ½a, n , and g is convex univalent in Δ such that ℏ ≺ g and y ≺ g ; then

The Results
In this section, we illustrate our main results concerning the class Σ α ℘ ðσ, pÞ for some special pðζÞ, ζ ∈ Δ.
Proof. Since then we obtain the next double inequality Thus, Lemmas 7 and 8 imply the desired assertion.

Theorem 12.
Let p be a univalent convex function in Δ such that pð0Þ = 0 and Proof. By the definition of ½Q m α ψðζÞ and ½L m α ψðζÞ, clearly we have ½A m α ψðζÞ ∈ Σ ℘ : Hence, a direct application of Lemma 9, we obtain the result.

Inclusion Properties.
In this part, we deal with the inclusion properties.
where dς conforms the probability measure on the unit circle |τ | = 1 and But, Y μ,ν ðζÞ is convex in Δ ; then we have Thus, ψ × f ∈ Σ α ℘ ðσ, pÞ: 2.3. Fekete-Szegö Inequality. In this section, we obtain the Fekete-Szegö relation coefficient estimates for the class Σ α ℘ ðσ, pÞ: Let Ω be the class of functions of the form in the open unit disk Δ satisfying |ϖðzÞ | <1: To prove our results, we need the following lemma.