Theory Of Differential Subordination Research Articles

Generalization of Jack’s Lemma for Functions With Fixed Initial Coefficient and Its Applications

In this paper, by using the theory of differential subordination, we will generalize Jack’s lemma for functions with fixed initial coefficient. Then extensions of the well-known open-door lemma for analytic and meromorphic functions with fixed initial coefficient are given. Also, we are considering some applications of the extension of Jack’s lemma. Keywords: Analytic functions, differential subordination, fixed initial coefficient, meromorphic functions, Nunokawa’s lemma, open-door lemma.

First-Order Differential Subordinations and Their Applications

In this paper, we consider some relations related to the representations of starlike and convex functions, and obtain some sufficient conditions for starlike and convex functions by using the theory of differential subordination. Actually, we generalize a result by Suffridge for analytic functions with missing coefficients and then we apply that generalization for obtaining the different methods to the implications of starlike or convex functions. Our results generalize and improve the previous results in the literature.

On New Symmetric Schur Functions Associated with Integral and Integro-Differential Functional Expressions in a Complex Domain

The symmetric Schur process has many different types of formals, such as the functional differential, functional integral, and special functional processes based on special functions. In this effort, the normalized symmetric Schur process (NSSP) is defined and then used to determine the geometric and symmetric interpretations of mathematical expressions in a complex symmetric domain (the open unit disk). To obtain more symmetric properties involving NSSP, we consider a symmetric differential operator. The outcome is a symmetric convoluted operator. Geometrically, studies are presented for the suggested operator. Our method is based on the theory of differential subordination.

SYMMETRICALLY CONFORMABLE FRACTIONAL DIFFERENTIAL OPERATORS BY COMPUTATIONAL NUMERICAL MODELING WITH SPECIAL FUNCTION

The [Formula: see text]-convoluted operators related to the [Formula: see text]-Whittaker function, confluent hypergeometric function of the first kind, have been developed using the [Formula: see text]-symbol calculus in which this sort of calculus presents a generalization of the gamma function. [Formula: see text]-symbol fractional calculus is employed to generalize and extend many differential and integral operators of fractional calculus. Based on this premise, a new geometric formula for normalized functions in the symmetric domain known as the open unit disk using the conformable fractional differential operator has been presented in this study. Thus, our technique entails investigating the most well-known geometric properties of this new operator, such as the subordination features and coefficient bounds so that the theory of differential subordination can be adjusted accordingly. By means of this technique, numerical results have been investigated for the proposed method. To this end, a few prominent corollaries of our primary findings as standout instances have been pointed out based on the positivity of the solutions, computational and numerical analyses.

Multivalent Functions and Differential Operator Extended by the Quantum Calculus

We used the concept of quantum calculus (Jackson’s calculus) in a recent note to develop an extended class of multivalent functions on the open unit disk. Convexity and star-likeness properties are obtained by establishing conditions for this class. The most common inequalities of the proposed functions are geometrically investigated. Our approach was influenced by the theory of differential subordination. As a result, we called attention to a few well-known corollaries of our main conclusions.

Boundedness of analytic functions in the unit disc using the theory of differential subordination

By using the theory of differential subordination, we investigate differential inequalities, satisfied by an analytic function [Formula: see text] defined on the unit disc, that imply the boundedness of the function [Formula: see text]. Our results are applied to find sufficient conditions for the normalized analytic functions to be either starlike or close-to-convex.

Differential subordination for Janowski functions with positive real part

"Theory of differential subordination provides techniques to reduce differential subordination problems into verifying some simple algebraic condition called admissibility condition.We exploit the first order differential subordination theory to get several sufficient conditions for function satisfying several differential subordinations to be a Janowski function with positive real part. As applications, we obtain suffcient conditions for normalized analytic functions to be Janowski starlike functions."

Differential Subordination and Superordination Results Using Fractional Integral of Confluent Hypergeometric Function

Both the theory of differential subordination and its dual, the theory of differential superordination, introduced by Professors Miller and Mocanu are based on reinterpreting certain inequalities for real-valued functions for the case of complex-valued functions. Studying subordination and superordination properties using different types of operators is a technique that is still widely used, some studies resulting in sandwich-type theorems as is the case in the present paper. The fractional integral of confluent hypergeometric function is introduced in the paper and certain subordination and superordination results are stated in theorems and corollaries, the study being completed by the statement of a sandwich-type theorem connecting the results obtained by using the two theories.

Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function

The idea of inequality has been extended from the real plane to the complex plane through the notion of subordination introduced by Professors Miller and Mocanu in two papers published in 1978 and 1981. With this notion came a whole new theory called the theory of differential subordination or admissible functions theory. Later, in 2003, a particular form of inequality in the complex plane was also defined by them as dual notion for subordination, the notion of differential superordination and with it, the theory of differential superordination appeared. In this paper, the theory of differential superordination is applied to confluent hypergeometric function. Hypergeometric functions are intensely studied nowadays, the interest on the applications of those functions in complex analysis being renewed by their use in the proof of Bieberbach’s conjecture given by de Branges in 1985. Using the theory of differential superodination, best subordinants of certain differential superordinations involving confluent (Kummer) hypergeometric function are stated in the theorems and relation with previously obtained results are highlighted in corollaries using particular functions and in a sandwich-type theorem. An example is also enclosed in order to show how the theoretical findings can be applied.

New Conditions for Univalence of Confluent Hypergeometric Function

Since in many particular cases checking directly the conditions from the definitions of starlikeness or convexity of a function can be difficult, in this paper we use the theory of differential subordination and in particular the method of admissible functions in order to determine conditions of starlikeness and convexity for the confluent (Kummer) hypergeometric function of the first kind. Having in mind the results obtained by Miller and Mocanu in 1990 who used a,c∈R, for the confluent (Kummer) hypergeometric function, in this investigation a and c complex numbers are used and two criteria for univalence of the investigated function are stated. An example is also included in order to show the relevance of the original results of the paper.

Best Subordinant for Differential Superordinations of Harmonic Complex-Valued Functions

The theory of differential subordinations has been extended from the analytic functions to the harmonic complex-valued functions in 2015. In a recent paper published in 2019, the authors have considered the dual problem of the differential subordination for the harmonic complex-valued functions and have defined the differential superordination for harmonic complex-valued functions. Finding the best subordinant of a differential superordination is among the main purposes in this research subject. In this article, conditions for a harmonic complex-valued function p to be the best subordinant of a differential superordination for harmonic complex-valued functions are given. Examples are also provided to show how the theoretical findings can be used and also to prove the connection with the results obtained in 2015.

Simple Sufficient Subordination Conditions for Close-to-Convexity

Using several applications of the theory of differential subordination we obtain sufficient conditions for usually normalized analytic functions to belong to certain subclasses of close-to-convex functions and close-to-convex functions of order α .

Some Special Differential Subordinations

For an analytic function $p$ satisfying $p(0)=1$, we obtain sharp estimates on $\beta$ such that the first order differential subordination $p(z)+\beta zp'(z)\prec \mathcal{P}(z)$ or $1+\beta zp'(z)/p^{j}(z)\prec \mathcal{P}(z)$, $(j=0,1,2)$ implies $p(z)\prec \mathcal{Q}(z)$ where $\mathcal{P}$ and $\mathcal{Q}$ are Caratheodory functions. The key tools in the proof of main results are the theory of differential subordination and some properties of hypergeometric functions. Further, these subordination results immediately give sufficient conditions for an analytic function $f$ to be in various well-known subclasses of starlike functions.

Differential Subordinations for Nonanalytic Functions

In the paper by Mocanu (1980), Mocanu has obtained sufficient conditions for a function in the classesC1(U), respectively, andC2(U)to be univalent and to mapUonto a domain which is starlike (with respect to origin), respectively, and convex. Those conditions are similar to those in the analytic case. In the paper by Mocanu (1981), Mocanu has obtained sufficient conditions of univalency for complex functions in the classC1which are also similar to those in the analytic case. Having those papers as inspiration, we try to introduce the notion of subordination for nonanalytic functions of classesC1andC2following the classical theory of differential subordination for analytic functions introduced by Miller and Mocanu in their papers (1978 and 1981) and developed in their book (2000). LetΩbe any set in the complex planeC, letpbe a nonanalytic function in the unit discU,p∈C2(U),and letψ(r,s,t;z):C3×U→C. In this paper, we consider the problem of determining properties of the functionp, nonanalytic in the unit discU, such thatpsatisfies the differential subordinationψ(p(z),Dp(z),D2p(z)-Dp(z);z)⊂Ω⇒p(U)⊂Δ.

Subordination and superordination properties of multivalent functions defined by certain integral operator

For normalized multivalent functions, we study different applications of the theory of differential subordination and superordination, that are connected to a recent linear operator defined by Liu and Owa.

An Application of Differential Subordination

We apply the general theory of differential subordination to obtain certian interesting criteria for -valent starlikeness and strong starlikeness. Some applications of these results are also discussed.

Double integral starlike operators†

The authors use the theory of differential subordinations to determine conditions on the kernel function W so that the function defined by is a starlike function.

Starlikeness of the Libera transform of functions with bounded turning

Conditions are determined for starlikeness of the Libera transform of functions with bounded turning. The results of Miller and Mocanu [S.S. Miller, P.T. Mocanu, Libera transform of functions with bounded turning, J. Math. Anal. Appl. 276 (2002) 90–97] are improved by using the theory of differential subordinations.

Briot–Bouquet differential superordinations and sandwich theorems

Briot–Bouquet differential subordinations play a prominent role in the theory of differential subordinations. In this article we consider the dual problem of Briot–Bouquet differential superordinations. Let β and γ be complex numbers, and let Ω be any set in the complex plane C. The function p analytic in the unit disk U is said to be a solution of the Briot–Bouquet differential superordination if Ω ⊂ { p ( z ) + z p ′ ( z ) β p ( z ) + γ | z ∈ U } . The authors determine properties of functions p satisfying this differential superordination and also some generalized versions of it. In addition, for sets Ω 1 and Ω 2 in the complex plane the authors determine properties of functions p satisfying a Briot–Bouquet sandwich of the form Ω 1 ⊂ { p ( z ) + z p ′ ( z ) β p ( z ) + γ | z ∈ U } ⊂ Ω 2 . Generalizations of this result are also considered.