Maximum Modulus Theorem Research Articles

L^q inequalities for polynomials with restricted zeros concerning the growth

If Pn denotes the class of polynomials of degree at most n, then for P ∈ Pn, a consequence of Maximum Modulus theorem yields for R > 1, || P(R, )||∞ < Rn || P ||∞. Various generalizations and refinements of this result are available in literature. In this paper, we consider a general class of polynomials Pn, μ, 1 < μ < n, with restriction on zeros in a specific way and obtain Zygmund-type inequalities concerning the growth of polynomials. Besides obtaining a refinement of a result due to Aziz and Shah, we improve a result of Aziz and Rather.

A Note on Location of the Zeros of Quaternionic Polynomials

The purpose of this paper is to investigate the extensions of the classical Eneström-Kakeya theorem and its various generalizations concerning the distribution of zeros of polynomials from the complex to the quaternionic setting. Using a maximum modulus theorem and the zero set structure in the recently published theory of regular functions and polynomials of a quaternionic variable, we construct new bounds of the Eneström-Kakeya type for the zeros of these polynomials. The obtained results for this subclass of polynomials and slice regular functions give generalizations of a number of results previously reported in the relevant literature.

On the Eneström-Kakeya theorem for quaternionic polynomial

The main purpose of this paper is to extend various results of Enestr?m-Kakeya type from the complex to quaternionic setting by virtue of a maximum modulus theorem and the structure of the zero sets in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our findings generalise several newly proven conclusions concerning the distribution of zeros of a quaternionic polynomial.

On the zeros of a quaternionic polynomial with restricted coefficients

In this paper, we are concerned with the problem of locating the zeros of regular polynomials of a quaternionic variable with restricted quaternionic coefficients. We derive new bounds of Eneström-Kakeya type for the zeros of these polynomials by virtue of a maximum modulus theorem and the structure of the zero sets established in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our results generalize some recently proved results about the distribution of zeros of a quaternionic polynomial.

On The Two-Dimensional Stokes Problem in Exterior Domains: The Maximum Modulus Theorem

The two-dimensional Stokes IBVP on (0,T)times Omega is investigated under the assumptions that Omega subset {mathbb {R}}^2 is a smooth exterior domain, the initial datum v_0 belongs to L^infty (Omega ) and (v_0,nabla phi )=0 for all phi in L^1_{ell oc}(Omega ) with nabla phi in L^1(Omega ). The well-posedeness in L^infty ((0,T)times Omega ) and the maximum modulus theorem are achieved, in particular one deduces that the Stokes semigroup on L^infty (Omega ) is a bounded analytic semigroup.

Some Inequalities for the Maximum Modulus of Rational Functions

For a polynomialpzof degreen, it follows from the maximum modulus theorem thatmaxz=R≥1pz≤Rnmaxz=1pz. It was shown by Ankeny and Rivlin that ifpz≠0forz<1, thenmaxz=R≥1pz≤Rn+1/2maxz=1pz. In 1998, Govil and Mohapatra extended the above two inequalities to rational functions, and in this paper, we study the refinements of these results of Govil and Mohapatra.

Paraunitary matrices, entropy, algebraic condition number and Fourier computation

The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey's Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time O(nlog⁡n). From a lower bound perspective, relatively little is known. Ailon shows in 2013 an Ω(nlog⁡n) bound for computing the normalized Fourier Transform assuming only unitary operations on two coordinates are allowed at each step, and no extra memory is allowed. In 2014, Ailon then improved the result to show that, in a κ-well conditioned computation, Fourier computation can be sped up by no more than O(κ). The main conjecture is that Ailon's result can be exponentially improved, in the sense that κ-well condition cannot admit ω(log⁡κ) speedup.The main result here is that ‘algebraic’ κ-well condition cannot admit ω(κ) speedup. One equivalent definition of algebraic condition number is related to the degree of polynomials naturally arising as the computation evolves. Using the maximum modulus theorem from complex analysis, we show that algebraic condition number upper bounds standard condition number, and equals it in certain cases. Algebraic condition number is an interesting measure of numerical computation stability in its own right, and provides a novel computational lens. Moreover, based on evidence from other recent related work, we believe that the approach of algebraic condition number has a good chance of establishing an algebraic version of the main conjecture.

Global $L^r$-estimates and regularizing effect for solutions to the $p(t,x)$-Laplacian systems

We consider the initial boundary value problem for the $p(t,x)$-Laplacian system in a bounded domain $\Omega$. If the initial data belongs to $L^{r_0}$, $r_0\geq 2$, we prove a global $L^{r_0}(\Omega)$-regularity result uniformly in $t>0$ that, in the particular case ${r_0}=\infty$, gives a maximum modulus theorem. Under the assumption $p_-=\inf p(t,x)>\frac{2n} {n+r_0}$, we also study $L^{r_0}-L^{r}$ estimates for the solution, for $r\geq r_0$.

The Schwarz lemma in Clifford analysis

In this paper, we first give the Cauchy type integral representation for harmonic functions in the Clifford analysis framework, and by using integral representations for harmonic functions in Clifford analysis, the Poisson integral formula for harmonic functions is represented. As its application, the mean value theorems and the maximum modulus theorem for Clifford-valued harmonic functions are presented. Second, some properties of Möbius transformations are given, and a close relation between the monogenic functions and Möbius transformations is shown. Finally, by using the integral representations for harmonic functions and the properties of Möbius transformations, Schwarz type lemmas for harmonic functions and monogenic functions are established.

On the Stokes problem in exterior domains: The maximum modulus theorem

We study the Stokes initial boundary value problem, in$(0,T) \times Ω$, where $Ω \subseteq \mathbb{R}^n$, $n\geq3$, is an exteriordomain, assuming that the initial data belongs to $L^\infty(Ω)$and has null divergence in weak sense. We prove the maximum modulustheorem for the corresponding solutions. Crucial for the proof ofthis result is the analogous one proved by Abe-Giga for boundeddomains. Our proof is developed by duality arguments and employingthe semigroup properties of the resolving operator defined on$L^1(Ω)$. Our results are similar to the ones proved by Solonnikovby means of the potential theory.

An Extention of the Hadamard-Type Inequality for a Convex Function Defined on Modulus of Complex Integral Functions

In this paper we extend the Hadamard's type inequalities for convex functions defined on the modulus of integral functions in complex field. Firstly, by using the Principal of maximum modulus theorem we show that $M(r)$ and $lnM(r)$ are continuous and convex functions for any non-negative values of $r$. Finally we derive two inequalities analogous to well known Hadamard's inequality by using elementary analysis.

On hypermonogenic functions

We research a function theory in higher dimensions based on the hyperbolic metric . The complex numbers are extended by the Clifford algebra Cl 0,n generated by the anti-commutating elements e i satisfying . In 1992, H. Leutwiler noticed that the power function (x 0 + x 1 e 1 + ··· + x n e n ) m is the generalized conjugate gradient of the functions . In the complex field (n = 1) this function h is harmonic in the usual sense, but in the higher dimensional case it is harmonic with respect to the Laplace–Beltrami operator with respect to the Riemannian hyperbolic metric . He started to study these type of functions, called H-solutions, that include positive and negative powers and elementary functions. Their total Clifford algebra valued generalizations, called hypermonogenic functions, are defined by H. Leutwiler and the first author in 2000. The integral formula has been proved by the first author. In this article, we present a simple way to find hyperbolic harmonic functions depending on the hyperbolic distance. We use these functions to determine a better presentation of the kernel, that is surprisingly the shifted Euclidean Cauchy kernel. We prove a power series expansion of hypermonogenic functions and present a version of the Maximum Modulus theorem.

A maximum modulus theorem for the Oseen problem

Classical solutions of the Oseen problem are studied on an exterior domain Ω with Ljapunov boundary in R3. It is proved a maximum modulus estimate of the following form: If \({{\bf u}\in C^2(\Omega)^3\cap C^0(\overline \Omega)^3}\) and \({p \in C^1(\Omega ), -\Delta {\bf u}+2\lambda \partial_1 {\bf u}+\nabla p=0, \nabla \cdot {\bf u}=0}\) in Ω, and if \({|{\bf u}| \le M}\) on \({\partial \Omega , \limsup |{\bf u}({\bf x})|\le M}\) as \({|{\bf x}|\to \infty }\) , then \({|{\bf u}({\bf x})|\le c M}\) in Ω. Here the constant c depends only on Ω and λ.

An introduction to complex analysis

Part 1 The complex plane: complex numbers open and closed sets in the complex plane limits and continuity. Part 2 Holomorphic function and power series: complex power series elementary functions. Part 3 Prelude to Cauchy's theorem: paths integration along paths connectedness and simple connectedness properties of paths and contours. Part 4 Cauchy's theorem: Cauchy's theorem, level I and II logarithms, argument and index Cauchy's theorem revisited. Part 5 Consequences of Cauchy's theorem: Cauchy's formulae power series representation zeros of holomorphic functions the maximum-modulus theorem. Part 6 Singularities and multifunctions: Laurent's theorem singularities meromorphic functions multifunctions. Part 7 Cauchy's residue theorem: counting zeroes and poles claculation of residues estimation of integrals. Part 8 Applications of contour integration: improper and principal-values integrals integrals involving functions with a finite number of poles and infinitely many poles deductions from known integrals integrals involving multifunctions evaluation of definite integrals. Part 9 Fourier and Laplace tranforms: the Laplace tranform - basic properties and evaluation the inversion of Laplace tranforms the Fourier tranform applications to differential equations proofs of the Inversion and Convolution theorems. Part 10 Conformal mapping and harmonic functions: circles and lines revisited conformal mapping mobius tranformations other mappings - powers, exponentials, and the Joukowski transformation examples on building conformal mappings holomorphic mappings - some theory harmonic functions.

A Phragmén - Lindelöf principle for slice regular functions

The celebrated 100-year old Phragmen-Lindelof principle is a far reaching extension of the maximum modulus theorem for holomorphic functions of one complex variable. In some recent papers there has been a resurgence of interest in principles of this type for functions of a hypercomplex variable and for solutions of suitable partial differential equations. In the present article we obtain a Phragmen-Lindelof principle for slice regular functions, a class of quaternion-valued functions of a quaternionic variable which has been recently introduced.

On the maximum modulus theorem for the steady-state Navier–Stokes equations in Lipschitz bounded domains

We prove that the steady-state Navier–Stokes problem in a Lipschitz bounded three-dimensional domain has a solution which satisfies a maximum modulus estimate, provided the fluxes of the boundary datum through the boundary of any interior connected component are less then a computable positive constant.

Generalized integral representations for functions with values in C(V 3,3)

By using the solution to the Helmholtz equation Δu − λu = 0 (λ ≥ 0), the explicit forms of the so-called kernel functions and the higher order kernel functions are given. Then by the generalized Stokes formula, the integral representation formulas related with the Helmholtz operator for functions with values in C(V3,3) are obtained. As application of the integral representations, the maximum modulus theorem for function u which satisfies Hu = 0 is given.

“Pointwise Asymptotic Stability of Steady Fluid Motions”

We study pointwise asymptotic stability of steady incompressible viscous fluids. The region of the motion is bounded. Our results of stability are based on the maximum modulus theorem that we prove for solutions of the Navier–Stokes equations. The asymptotic stability is based on a variational formulation. Since the region of the motion is bounded, the time decay is of exponential type. Of course suitable assumptions are made about the smallness of the size of the uniform norm of the perturbations at the initial data. With no restrictions, we are able only to prove an existence theorem of the perturbation local in time.

Converse of the Maximum Modulus Theorem and Rings of Continuous Complex Functions

It is shown among others that if D is a region (öpen connected set) in the complex plane and C(D) the algehra of continuous complex functions on Dwith the property that for every m em ber/c C (D) and every closed disc M7 in D, ||/||^ r = ||/ ||g ^ then any two such regions are conformally equivalent. Moreover ev ery /is analytic.

A Cauchy's Integral Formula for Functions with Values in a Universal Clifford Algebra and its Applications

In this paper, we obtain Cauchy's integral formula on certain distinguished boundary for functions with values in a universal Clifford algebra, which is similar to the classical Cauchy's integral formula on the distinguished boundary of polycylinder for several complex variables. By using it, both the mean value theorem and the maximum modulus theorem are given.