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The thermodynamical formalism has been developed by the authors for a very general class of transcendental meromorphic functions. A function In the present manuscript we first improve upon our earlier paper in providing a systematic account of the thermodynamical formalism for such a meromorphic function Then we provide various, mainly geometric, applications of this theory. Indeed, we examine the finer fractal structure of the radial (in fact non-escaping) Julia set by developing the multifractal analysis of Gibbs states. In particular, the Bowen’s formula for the Hausdorff dimension of the radial Julia set from our earlier paper is reproved. Moreover, the multifractal spectrum function is proved to be convex, real-analytic and to be the Legendre transform conjugate to the temperature function. In the last chapter we went even further by showing that, for a analytic family satisfying a symmetric version of the growth condition (1.1) in a uniform way, the multifractal spectrum function is real-analytic also with respect to the parameter. Such a fact, up to our knowledge, has not been so far proved even for hyperbolic rational functions nor even for the quadratic family

We define new classes of meromorphic p-valent convex functions, respectively, meromorphic close-to-convex functions, by using an extension of Wanas operator in order to study the preservation properties of these classes, when a well-known integral operator is used. We find the conditions which allow this operator to preserve the classes mentioned above, and we will remark the symmetry between these classes.

The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth. In this present paper we consider the normal matrix model with cubic plus linear potential. In order to regularize the model, we follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain We also study in detail the mother body problem associated to To construct the mother body measure, we define a quadratic differential Following previous works of Bleher & Kuijlaars and Kuijlaars & López, we consider multiple orthogonal polynomials associated with the normal matrix model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain strong asymptotic formulas for these polynomials. Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of the associated

The theory of entire and meromorphic functions is a very important area of complex analysis. This monograph aims to expand the discussion about some growth properties of integer translated composite entire and meromorphic functions on the basis of their (p,q,t)L -order and (p,q,t)L -type. This book presents six chapters. Chapter 1 introduces the reader to the preliminary definitions and notations. Chapter 2 and Chapter 3 discuss some results related to (p; q; t) L-th order and (p; q; t)L-th lower order of composite entire and meromorphic functions on the basis of their integer translation. Chapter 4 establishes some relations of integer translated composite entire and meromorphic functions based on their (p; q; t) L-th type and (p; q; t) L-th weak type. Chapter 5 deals with some results about (p; q; t) L-th order and (p; q; t) L-th type of composite entire and meromorphic functions on the basis of their integer translation. Chapter 6 focuses on some results about (p; q; t) L-th order and (p; q; t) L-th type of composite entire and meromorphic functions on the basis of their integer translation. This monograph will be very helpful for postgraduates, researchers, and faculty members interested in value distribution theorems in complex mathematical analysis.

Abstract-Let А\" = {г : z € C,1 < |z| < +00), and consider the class Mx (и, A, ф) of meromorphic bi-univalent functions defined т 4\". This work focuses on deriving estimates for the coefficients |bo|, |b1| and |b2| of functions in Mx (и, A, &), utilizing the properties of meromorphic functions. The findings presented here refine or extend certain results established by earlier researchers.

This paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set.

The tropical analogue of the lemma on the logarithmic derivative is generalised for noncontinuous tropical meromorphic functions, that is, piecewise linear functions that may have discontinuities. In addition, two Borel type results are generalised for piecewise continuous functions. With the generalisation of the tropical analogue of the lemma on the logarithmic derivative, several tropical analogues of Clunie and Mohon’ko type results are also automatically generalised for noncontinuous tropical meromorphic functions.

In this article, we construct filling disks for meromorphic functions of order zero and that way we prove the existence of Borel directions of these functions. In the latter part of this article, we demonstrate the existence of filling disks using the Borel direction of meromorphic functions.

Let ${\\mathbb D}$ be the open unit disk, and let $\\mathcal {A}(p)$ be the class of functions f that are holomorphic in ${\\mathbb D}\\backslash \\{p\\}$ with a simple pole at $z=p\\in (0,1)$ , and $f'(0)\\neq 0$ . In this article, we significantly improve lower bounds of the Bloch and the Landau constants for functions in ${\\mathcal A}(p)$ which were obtained in Bhowmik and Sen (2023, Monatshefte für Mathematik, 201, 359–373) and conjecture on the exact values of such constants.

We consider a newly introduced integral operator that depends on an analytic normalized function and generalizes many other previously studied operators. We find the necessary conditions that this operator has to meet in order to preserve convex meromorphic functions. We know that convexity has great impact in the industry, linear and non-linear programming problems, and optimization. Some lemmas and remarks helping us to obtain complex functions with positive real parts are also given.