Department of Mathematics

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

In this article, we first propose the modified Hannan–Rissanen Method for estimating the parameters of autoregressive moving average (ARMA) process with symmetric stable noise and symmetric stable generalized autoregressive conditional heteroskedastic (GARCH) noise. Next, we propose the modified empirical characteristic function method for the estimation of GARCH parameters with symmetric stable noise. Further, we show the efficiency, accuracy and simplicity of our methods with Monte-Carlo simulation. Finally, we apply our proposed methods to model the financial data. © 2021 Informa UK Limited, trading as Taylor & Francis Group.

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Journal of Applied Statistics

Estimation of the parameters of symmetric stable ARMA and ARMA–GARCH models

The aim of this paper is to establish some properties of solutions to the Dirichlet–Neumann problem: (∂z∂z¯)2w=g in the unit disc D, w= γ and ∂ν∂z∂z¯w= γ on T (the unit circle), 12πi∫Twζζ¯(ζ)dζζ=c, where ∂ν denotes differentiation in the outward normal direction. More precisely, we obtain Schwarz–Pick type inequalities and Landau type theorem for solutions to the Dirichlet–Neumann problem. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

Computational Methods and Function Theory

Schwarz–Pick and Landau Type Theorems for Solutions to the Dirichlet–Neumann Problem in the Unit Disk

Let f= P[F] denote the Poisson integral of F in the unit disk D with F being absolutely continuous in the unit circle T and F˙ ∈ Lp(T) , where F˙(eit)=ddtF(eit) and p≥ 1. Recently, the author in Zhu (J Geom Anal, 2020) proved that (1) if f is a harmonic mapping and 1 ≤ p&lt; 2 , then fz and fz¯¯ ∈ Bp(D) , the classical Bergman spaces of D [12, Theorem 1.2]; (2) if f is a harmonic quasiregular mapping and 1 ≤ p≤ ∞, then fz, fz¯¯ ∈ Hp(D) , the classical Hardy spaces of D [12, Theorem 1.3]. These are the main results in Zhu (J Geom Anal, 2020). The purpose of this paper is to generalize these two results. First, we prove that, under the same assumptions, [12, Theorem 1.2] is true when 1 ≤ p&lt; ∞. Also, we show that [12, Theorem 1.2] is not true when p= ∞. Second, we demonstrate that [12, Theorem 1.3] still holds true when the assumption f being a harmonic quasiregular mapping is replaced by the weaker one f being a harmonic elliptic mapping. © 2021, Mathematica Josephina, Inc.

Journal of Geometric Analysis

Remarks on ‘Norm Estimates of the Partial Derivatives for Harmonic Mappings and Harmonic Quasiregular Mappings’

There are a number of articles which deal with Bohr's phenomenon whereas only a few papers appeared in the literature on Rogosinski's radii for analytic functions defined on the unit disk |z|<1. In this article, we introduce and investigate Bohr-Rogosinski's phenomenon for analytic functions defined for |z|<1 in a general setting. In addition we discuss and derive Bohr-Rogosinski's radii for Cesáro operators on the space of bounded analytic functions. Finally, we also obtain the Bohr-Rogosinski's radius for a class of subordinations. All the results are proved to be sharp. © 2020 Elsevier Inc.

Journal of Mathematical Analysis and Applications

Bohr–Rogosinski phenomenon for analytic functions and Cesáro operators

In Li et al. (J Aust Math Soc 97:383–390, 2014), the authors discussed two classes of mappings in Banach spaces, which are φ-quasiisometric mappings in the distance ratio metric (briefly, φ-DRQI mappings) and fully φ-quasiisometric mappings in the distance ratio metric (briefly, φ-FDRQI mappings), where φ: [0 , ∞) → [0 , ∞) denotes a quasiisometric control function. In this paper, we continue this discussion by introducing a new class of mappings, that is, φ-PDRQI mappings. The purpose of this paper is twofold. Let f be a homeomorphism between two proper domains in a Banach space. First, we prove that f being φ-PDRQI is equivalent to f being φ-FDRQI. Then, as an application of the obtained equivalent relation, we show that f being φ-DRQI, f being φ-PDRQI and f being φ-FDRQI are equivalent to each other when the quasiisometric control functions have the form φ(t) = Mt. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.

Monatshefte fur Mathematik

Quasiisometric mappings in the distance ratio metric in Banach spaces

In this paper, we first determine Bohr’s inequality for the class of harmonic mappings f= h+ g¯ in the unit disk D, where either both h(z)=∑n=0∞apn+mzpn+m and g(z)=∑n=0∞bpn+mzpn+m are analytic and bounded in D, or satisfies the condition | g′(z) | ≤ d| h′(z) | in D\  0  for some d∈ [0 , 1] and h is bounded. In particular, we obtain Bohr’s inequality for the class of harmonic p-symmetric mappings. Also, we investigate the Bohr-type inequalities of harmonic mappings with a multiple zero at the origin and that most of results are proved to be sharp. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.

Mediterranean Journal of Mathematics

Bohr-Type Inequalities for Harmonic Mappings with a Multiple Zero at the Origin

In this paper, we prove that the inversions in metric spaces are coarsely bilipschitz continuous with respect to distance ratio metric. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.

Results in Mathematics

Coarsely Bilipschitz Continuity of Inversions with Respect to Distance Ratio Metrics in Metric Spaces

In this paper, we first establish a version of multidimensional analogues of the refined Bohr's inequality. Then we establish two versions of multidimensional analogues of improved Bohr's inequality with initial coefficient being zero. Finally we establish two versions of multidimensional analogues of improved Bohr's inequality with the initial coefficient being replaced by absolute value of the function, and to prove that most of the results are sharp. © 2021 American Mathematical Society.

Proceedings of the American Mathematical Society

Multidimensional analogues of refined bohr's inequality

I. M. Milin proposed, in his 1971 paper, a system of inequalities for the logarithmic coefficients of normalized univalent functions on the unit disk of the complex plane. This is known as the Milin conjecture and implies the Robertson conjecture which in turn implies the Bieberbach conjecture. In 1984, Louis de Branges settled the long-standing Bieberbach conjecture by showing the Milin conjecture. Recently, O. Roth proved an interesting sharp inequality for the logarithmic coefficients based on the proof by de Branges. In this paper, following Roth's ideas, we will show more general sharp inequalities with convex sequences as weight functions. By specializing the sequence, we can obtain an abundant number of sharp inequalities on logarithmic coefficients, some of which are provided in Appendix. We also consider the inequality with the help of de Branges system of linear ODE for non-convex sequences where the proof is partly assisted by computer. Also, we apply some of those inequalities to improve previously known results. © 2020 Elsevier Masson SAS

Bulletin des Sciences Mathematiques

Sharp inequalities for logarithmic coefficients and their applications

In this paper, we study the Bohr phenomenon for functions that are defined on a general simply connected domain of the complex plane. We improve known results of R. Fournier and St. Ruscheweyh for a class of analytic functions. Furthermore, we examine the case where a harmonic mapping is defined in a disk containing D and obtain a Bohr type inequality. © 2021, Springer Nature Switzerland AG.