Let S(p) represent the collection of meromorphic univalent functions f in the unit disc D which possess a simple pole at z=p(0<p<1) and meet the normalization f(0)=f?(0)-1=0. In this article, we determine bounds for Hermitian Toeplitz determinants whose entries are the Taylor coefficients of functions in S(p). Furthermore, we derive bounds for Hermitian Toeplitz determinants for two specific subclasses of S(p). © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2025

We study the class which includes functions f that are meromorphic in the unit disk and have a simple pole at for some with the normalization. We establish a sufficient condition for functions in this class to be univalent. Making use of this condition, we introduce a subfamily consisting of univalent functions satisfying a certain differential inequality in . Next, we obtain a representation formula for such functions. Additionally, we establish necessary and sufficient conditions on the coefficients for functions of the form belong to this class. Furthermore, we determine sharp upper bounds for all. Finally, we establish a sharp estimate for the Fekete-Szego functional associated with the newly introduced subclass

We obtain bounds for certain functionals defined on a class of meromorphic functions in the unit disc of the complex plane with a nonzero simple pole. These bounds are sharp in a certain sense. We also discuss possible applications of this result. Finally, we generalise the result to meromorphic functions with more than one simple pole.

Let Co(p) be the class of all functions f defined in the unit disc D having a simple pole at z = p where 0 < p < 1 and analytic in D \ {p} with f (0) = 0 = f ?(0) ? 1 such that f maps D onto a domain whose complement with respect to the extended complex plane is a bounded convex set. These functions are called concave univalent functions. Each f ? Co(p) has the following Taylor expansion: (Farmula Presented) In this article, we first determine the regions of variability of the difference of successive coefficients (an+1 ? an) for n ? 3 . We also find sharp upper bounds of the Toeplitz determinants, the entries of which are the Taylor coefficients of functions in Co(p) . © 2024 Element D.O.O.. All rights reserved.

Let V(?) be the class of all functions f defined on the open unit disc D of the complex plane having simple pole at z= p, p? (0 , 1) and analytic in D\ { p} satisfying the normalizations f(0) = 0 = f(0) - 1 such that | (z/ f(z)) f(z) - 1 | < ? for z? D, ?? (0 , 1]. In this article, we obtain sharp bounds of the Zalcman and the generalized Zalcman functionals for functions in V(?) for some indices of these functionals. As consequences of the obtained results, we pose the Zalcman-like coefficient conjectures for this class of functions. In addition, we estimate bound for the generalised Fekete–Szegö functional along with bounds of the second- and the third-order Hankel determinants for this class of functions.

Prism networks belong to generalized petersen graph topologies with planar and polyhedral properties. These networks are extensively used to study the complex networks in the context of computer science, biological networks, brain networks, and social networks. In this letter, the performance analysis of consensus algorithms over prism networks is investigated in terms of convergence time, network coherence, and communication delay. Specifically, in this letter, we first derive the explicit expressions for eigenvalues of Laplacian matrix over m -dimensional prism networks using spectral graph theory. Subsequently, the study of consensus metrics such as convergence time, maximum communication time delay, first order network coherence, and second order network coherence is performed. Our results indicate that the effect of noise and communication time-delay on the consensus dynamics in m -dimensional prism networks is minimal. The obtained results illustrate that the scale-free topology of m -dimensional prism networks along with loopy structure is responsible for strong robustness with respect to consensus dynamics in prism networks.

The distance matrix of a simple connected graph G is (Formula presented.) where d is the length of a shortest path between the ith and jth vertices of G. Eigenvalues of D(G) are called the distance eigenvalues of G. The m-generation n-prism graph or (m, n)-prism graph can be defined in an iterative way where (Formula presented.) -prism graph is an n-vertex cycle. In this paper, we first find the number of zero eigenvalues of the distance matrix of a (m, n)-prism graph. Next, we find some quotient matrix that contains m nonzero distance eigenvalues of a (m, n)-prism graph. Our next result gives the rest of the nonzero distance eigenvalues of a (m, n)-prism graph in terms of distance eigenvalues of a cycle. Finally, we find the characteristic polynomial of the distance matrix of a (m, n)-prism graph. Applying this result, we provide the explicit distance eigenvalues of a (Formula presented.) -prism graph.