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."

Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot Fractal Set

We present an analytic solvability of a class of Langevin differential equations (LDEs) in the asset of geometric function theory. The analytic solutions of the LDEs are presented by utilizing a special kind of fractal function in a complex domain, linked with the subordination theory. The fractal functions are suggested for the multi-parametric coefficients type motorboat fractal set. We obtain different formulas of fractal analytic solutions of LDEs. Moreover, we determine the maximum value of the fractal coefficients to obtain the optimal solution. Through the subordination inequality, we determined the upper boundary determination of a class of fractal functions holding multibrot function ϑ(z)=1+3κz+z3.

Subclasses of Multivalent Analytic Functions Associated with a q-Difference Operator

In this article we introduced and studied some inclusion properties for new subclasses of multivalent analytic functions defined by using the q-derivative operator. With the aid of the Jackson q-derivative we defined two new operators that generalize many other previously studied operators, and help us to define two new subclasses of functions with several interesting properties studied in this paper. The methods used for the proof of our results are special tools of the differential subordination theory of one-variable functions.

On the Jack lemma and its generalization

We present a new geometric approach to some problems in differential subordination theory and discuss the new results closely related to the Jack lemma.

The Meaning of Silence in Cyberspace

It has been argued in the literature on hate speech and subordination theory that even ordinary hate speakers who are not “figures of authority” in the conventional sense can possess the power or authority to subordinate (rank as inferior, deny rights and powers, or legitimate discrimination against) the targets of their hate speech in virtue of the fact that when witnesses to the hate speech remain silent they “license” or grant authority to the hate speaker. A typical example is when a hate speaker targets a victim on public transport and other passengers remain silent. The aim of this chapter is to examine the extent to which this account of licensing is applicable to online hate speech or cyberhate. More generally, the chapter explores whether the potentially distinctive nature of online communication changes the meaning of silence such that it becomes difficult to interpret silence in cyberspace as acquiescence, licensing, or complicity.

Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem

We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series expansions, but are defined as Fréchet analytic functions in all of the operator upper half plane. Furthermore, we do not have to assume that our state is tracial. Combining this new analytic theory of operator-valued free convolution with Anderson’s selfadjoint version of the linearization trick we are able to provide a solution to the following general random matrix problem: Let {X_{1}^{(N)},\dots,X_{n}^{(N)}} be selfadjoint {N\times N} random matrices which are, for {N\to\infty} , asymptotically free. Consider a selfadjoint polynomial p in n non-commuting variables and let {P^{(N)}} be the element {P^{(N)}=p(X_{1}^{(N)},\dots,X_{n}^{(N)})} . How can we calculate the asymptotic eigenvalue distribution of {P^{(N)}} out of the asymptotic eigenvalue distributions of {X_{1}^{(N)},\dots,X_{n}^{(N)}} ?

Some applications of first-order differential subordinations

In this work we present a new geometric approach to some problems in differential subordination theory. In the paper some sufficient conditions for function to be starlike or univalent or to be in the class of Carathéodory functions are obtained. We also discuss the new results closely related to the generalized Briot-Bouquet differential subordination.

Spatial Branching Processes and Subordination

We present a subordination theory for spatial branching processes. This theory is developed in three different settings, first for branching Markov processes, then for superprocesses and finally for the path-valued process called the Brownian snake. As a common feature of these three situations, subordination can be used to generate new branching mechanisms. As an application, we investigate the compact support property for superprocesses with a general branching mechanism.