New Symmetric Differential and Integral Operators Defined in the Complex Domain

Investigation of the theory of operators (differential, integral, mixed, convolution and linear) has been a capacity of apprehension for numerous scientists in all fields of mathematical sciences, such as mathematical physics, mathematical biology and mathematical computing. An additional definite field is the study of inequalities in the complex domain. Works’ review shows masses of studies created by the classes of analytic functions. The relationship of geometry and analysis signifies a very central feature in geometric function theory in the open unit disk. This fast development is directly connected to the existence between analysis, construction and geometric performance [1]. In 1983, Sàlàgean introduced his famous differential operator of normalized analytic functions in the open unit disk [2]. This operator is generalized and extended to many classes of univalent functions. It plays a significant tool to develop the geometric structure of many analytic functions by suggesting different classes. Later this operator has been generalized and motivated by many researchers, for example, the Al-Oboudi differential operator [3]. Recently, a new study is presented by using the Sàlàgean operator [4]. Our research is to formulate a new symmetric differential operator and its integral by utilizing the concept of the symmetric derivative of complex variables. This concept is an operation, extending the original derivative. Note that its practical use in the the symmetry models in math modeling remains open. For example, for application in mathematical physics it is critical to employ group analysis methods. Such methods enable methods for branching solutions construction using group symmetry [5,6].

We shall need the following basic definitions throughout this paper. A function $\varphi \in \Lambda$ is said to be univalent in $\mathbb{U}$ if it never takes the same value twice; that is, if $z_1\ne z_2$ in the open unit disk $\mathbb{U}=\{z\in \mathbb{C}:|z|<1\}$ then $\varphi \left(z_1\right)\ne \varphi \left(z_2\right)$ or equivalently, if $\varphi \left(z_1\right)=\varphi \left(z_2\right)$ then $z_1=z_2$. Without loss of generality, we can use the notion $\Lambda$ for our univalent functions taking the expansion

We let $\mathcal{S}$ denote the class of such functions $\varphi \in \Lambda$ that are univalent in $\mathbb{U}$.

A function $\varphi \in \mathcal{S}$ is said to be starlike with respect to origin in $\mathbb{U}$ if the linear segment joining the origin to every other point of $\varphi (z:|z|=r<1)$ lies entirely in $\varphi (z:|z|=r<1)$. In more picturesque language, the requirement is that every point of $\varphi (z:|z|=r<1)$ be visible from the origin. A function $\varphi \in \mathcal{S}$ is said to be convex in $\mathbb{U}$ if the linear segment joining any two points of $\varphi (z:|z|=r<1)$ lies entirely in $\varphi (z:|z|=r<1)$. In other words, a function $\varphi \in \mathcal{S}$ is said to be convex in $\mathbb{U}$ if it is starlike with respect to each and every of its points. We denote the class of functions $\varphi \in \mathcal{S}$ that are starlike with respect to origin by ${\mathcal{S}}^{*}$ and convex in $\mathbb{U}$ by $\mathcal{C}$.

Neatly linked to the classes ${\mathcal{S}}^{*}$ and $\mathcal{C}$ is the class $\mathcal{P}$ of all functions $\varphi$ analytic in $\mathbb{U}$ and having positive real part in $\mathbb{U}$ with $\varphi \left(0\right)=1$. In fact $f\in {\mathcal{S}}^{*}$ if and only if $z{\varphi }^{\prime }\left(z\right)/\varphi \left(z\right)\in \mathcal{P}$ and $\varphi \in \mathcal{C}$ if and only if $1+z{\varphi }^{\prime \prime }\left(z\right)/{\varphi }^{\prime }\left(z\right)\in \mathcal{P}$. In general, for $ϵ\in [0,1)$ we let $\mathcal{P}\left(ϵ\right)$ consist of functions $\varphi$ analytic in $\mathbb{U}$ with $\varphi \left(0\right)=1$ so that $\Re \left(\varphi \right(z\left)\right)>ϵ$${(}^{\prime }{\Re }^{\prime }$ represents to the real part) for all $z\in \mathbb{U}$. Note that $\mathcal{P}\left({ϵ}_2\right)\subset \mathcal{P}\left({ϵ}_1\right)\subset \mathcal{P}\left(0\right)\equiv \mathcal{P}$ for $0<{ϵ}_1<{ϵ}_2$ (e.g., see Duren [1]).

For functions $\varphi$ and $\psi$ in $\Lambda$ we say that $\varphi$ is subordinate to $\psi$, denoted by $\varphi \prec \psi$, if there exists a Schwarz function $\omega$ with $\omega \left(0\right)=0$ and $\left|\omega \right(z\left)\right|<1$ so that $\varphi \left(z\right)=\psi \left(\omega \right(z\left)\right)$ for all $z\in \mathbb{U}$ (see [7]). Evidently $\varphi \left(z\right)\prec \psi \left(z\right)$ is equivalent to $\varphi \left(0\right)=\psi \left(0\right)$ and $\varphi \left(\mathbb{U}\right)\subset \psi \left(\mathbb{U}\right).$ We request the following results, which can be located in [7].

Let $\varphi \in \Lambda ,$ taking the power series (1). For a function $\varphi \left(z\right)$ and a constant $\alpha \in [0,1],$ we formulate the SDO as follows:

It is clear that when $\alpha =1,$ we have Sàlàgean differential operator [2]${\mathcal{S}}^k\varphi \left(z\right)=z+{\sum }_{n=2}^{\infty }{n}^k{\phi }_n{z}^n.$ We may say that SDO (2) is the symmetric Sàlàgean differential operator in the open unit disk. In the same manner of the formula of Sàlàgean integral operator, we consume that for a function $\varphi \in \Lambda ,$ the symmetric integral operator ${\mathcal{J}}_{\alpha }^k$ satisfies

Similarly, when $\alpha =1,$ we have Sàlàgean integral operator [2], Remark 5. Furthermore, we conclude the relation ${\mathcal{M}}_{\alpha }^k\left({\mathcal{J}}_{\alpha }^k\varphi \left(z\right)\right)={\mathcal{J}}_{\alpha }^k\left({\mathcal{M}}_{\alpha }^k\varphi \left(z\right)\right)=\varphi \left(z\right).$

Next, we proceed to formulate a linear combination operator involving SDO and the Ruscheweyh derivative. For a function $\varphi \in \Lambda ,$ the Ruscheweyh derivative achieves the formula where the term $C_{k+n-1}^k$ is the combination coefficients. In this note, we introduce a new operator combining $R^k$ and ${\mathcal{M}}_{\alpha }^k$ as follows:

We shall deal with the following classes

Obviously, the subclass $S_0^{*}\left(h\right)={\mathcal{S}}^{*}\left(h\right).$

In this section, we utilize the above constructions of the symmetric operators to get some geometric fulfillment.

Next consequence result of Theorem 3 can be found in [9,11] respectively.

Next consequence result of Theorem 4 can be found in [8].

We inform the readers that in virtue of Noshiro-Warschawski Theorem (Duren [1], p. 47) if a function $\varphi$ is analytic in the simply connected complex domain $\mathbb{U}$ and $\Re \left\{{\varphi }^{\prime }\left(z\right)\right\}>0$ in $\mathbb{U}$ then $\varphi$ is univalent in $\mathbb{U}$ and in view of Kaplan’s Theorem (Duren [1], p. 48) such functions $\varphi$ is of bounded boundary rotation.

Motivated by this method, in the recent investigation we have presented new classes of univalent functions that connect to a symmetric differential operator in the open unit disk. We have obtained sufficient and necessary conditions in relation to these subclasses. Linear combinations, operator and other properties are also explored. For further research, we indicate to study the certain new classes related to other types of analytic functions such as meromorphic, harmonic and p-valent functions with respect to symmetric points associated with SDO.