Bohr Inequality Research Articles

On operator Bohr type inequalities

The purpose of this paper is to discuss inequalities related to operator versions of the classical Bohr inequality. We obtain refinements of some inequalities due to Cheung and Pecaric [J. Math. Anal. Appl. 323 (2006) 403–412] and Zhang [J. Math. Anal. Appl. 333 (2007) 1264–1271]. Moreover, we present two inequalities for multiple operators, which are similar to ones proposed by Chansangiam et al. [J. Math. Anal. Appl. 356 (2009) 525–536]. Mathematics subject classification (2010): 47A63.

Bohr's phenomenon for analytic functions and the hyperbolic metric

AbstractA link is established between Bohr's inequality for classes of analytic functions and the hyperbolic metric. The classes considered consist of analytic functions mapping the unit disk respectively into the right half‐plane, the slit region, and to the exterior of the unit disk.

Bohr's phenomenon in subordination and bounded harmonic classes

We first establish Bohr's inequality for the class of analytic functions subordinate to univalent functions. Then, we establish a Bohr's inequality for harmonic functions mapping D into D.

Generalizations of Bohr's inequality in Hilbert C*-modules

We present a new operator equality in the framework of Hilbert C*-modules. As a consequence, we get an extension of the Euler–Lagrange type identity in the setting of Hilbert bundles as well as several generalized operator Bohr's inequalities due to O. Hirzallah, W.-S. Cheung, J.E. Pečarić and F. Zhang.

Matrix order in Bohr inequality for operators

The classical Bohr inequality says that $|a+b|^2\leq p|a|^2+q|b|^2$ for all scalars $a, \ b$ and positive $p,q$ with $\frac 1p + \frac 1q =1.$ The equality holds if and only if $(p-1)a=b.$ Several authors discussed operator version of Bohr inequality. In this paper, we give a unified proof to operator generalizations of Bohr inequality. One viewpoint of ours is a matrix inequality, and the other is a generalized parallelogram law for absolute value of operators, i.e., for operators $A$ and $B$ on a Hilbert space and $t\neq0$, $$|A-B|^2+\frac{1}{t}|tA+B|^2=(1+t)|A|^2+(1+\frac{1}{t})|B|^2.$$

Generalizations of Bohr inequality for Hilbert space operators

Let B ( H ) be the space of all bounded linear operators on a complex separable Hilbert space H . Bohr inequality for Hilbert space operators asserts that for A , B ∈ B ( H ) and p , q > 1 real numbers such that 1 / p + 1 / q = 1 , | A + B | 2 ⩽ p | A | 2 + q | B | 2 with equality if and only if B = ( p − 1 ) A . In this paper, a number of generalizations of Bohr inequality for operators in B ( H ) are established. Moreover, Bohr inequalities are extended to multiple operators and some related inequalities are obtained. The results in this paper generalize results known so far. The idea of transforming problems in operator theory to problems in matrix theory, which are easy to handle, is the key role.

Multivariable Bohr inequalities

Operator-valued multivariable Bohr type inequalities are obtained for: (i) a class of noncommutative holomorphic functions on the open unit ball of , generalizing the analytic functions on the open unit disc; (ii) the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra ; (iii) a class of noncommutative selfadjoint harmonic functions on the open unit ball of , generalizing the real-valued harmonic functions on the open unit disc; (iv) the Cuntz-Toeplitz algebra , the reduced (resp. full) group -algebra (resp. ) of the free group with generators; (v) a class of analytic functions on the open unit ball of . The classical Bohr inequality is shown to be a consequence of Fejer's inequality for the coefficients of positive trigonometric polynomials and Haager- up-de la Harpe inequality for nilpotent operators. Moreover, we provide an inequality which, for analytic polynomials on the open unit disc, is sharper than Bohr's inequality.

On the Bohr inequality of operators

This paper is focused on the operator inequalities of the Bohr type. We will give a new and transparent proof for the operator Bohr inequality through an absolute value operator identity, show some related operator inequalities by means of 2 × 2 (block) operator matrices, and finally we will present a generalization of the operator Bohr inequality for multiple operators.

The Bohr inequality for ordinary Dirichlet series

We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if f(s) = P ∞=1 ann −s with k f k ∞ := supℜs>0 |f(s)| < ∞, then P ∞=1 |an|n −2 ≤ k f k ∞ and even slightly better, and P∞=1 |an|n −1/2 ≤ Ck f k ∞, C being an absolute constant.

Bohr's inequalities for Hilbert space operators

The classical Bohr's inequality states that | z + w | 2 ⩽ p | z | 2 + q | w | 2 for all z , w ∈ C and all p , q > 1 with 1 p + 1 q = 1 . In this paper, Bohr's inequality is generalized to the context of Hilbert space operators for all positive conjugate exponents p , q ∈ R . In particular, the parallelogram law is recovered and some other interesting operator inequalities and established.

A remark on Bohr's inequality

This paper studies the classical problem of determining the maximum growth of the Taylor series majorant of holomorphic functions with bound 1 in the unit disk, as a function of the radius of the Taylor series. We determine for the first time the asymptotic behavior of this maximum as the radius tends to 1, with a very small error term. The proof uses full asymptotic expansions of certain oscillatory integrals and the discrete Hilbert transform.

Non-commutative operator Bohr inequality

It is shown that if A, B, X are Hilbert space operators such that X⩾ γI, for the positive real number γ, and p, q>1 with 1/ p+1/ q=1, then | A− B| 2⩽ p| A| 2+ q| B| 2 with equality if and only if (1− p) A= B and γ|||| A− B| 2|||⩽||| p| A| 2 X+ qX| B| 2||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert–Schmidt operator, then ‖ δ A, B 2( X)‖ 2⩽‖ p| A| 2 X+ qX| B| 2‖ 2 with equality if and only if (1− p) AX= XB.

ON BOHR'S INEQUALITY

Bohr's inequality says that if $f(z) = \sum^{\infty}_{n = 0} a_n z^n$ is a bounded analytic function on the closed unit disc, then $\sum^{\infty}_{n = 0} \lvert a_n\rvert r^n \leq \Vert f\Vert_{\infty}$ for $0 \leq r \leq 1/3$ and that $1/3$ is sharp. In this paper we give an operator-theoretic proof of Bohr's inequality that is based on von Neumann's inequality. Since our proof is operator-theoretic, our methods extend to several complex variables and to non-commutative situations.We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher-dimensional domains, for the non-commutative disc algebra ${\mathcal A}_n$, and for the reduced (respectively full) group C*-algebra of the free group on $n$ generators.We also include an application to Banach algebras. We prove that every Banach algebra has an equivalent norm in which it satisfies a non-unital version of von Neumann's inequality.2000 Mathematical Subject Classification: 47A20, 47A56.

APPROXIMATION OF PERIODIC FUNCTIONS OF SEVERAL VARIABLES WITH BOUNDED MIXED DIFFERENCE

This paper studies questions concerning the approximation of functions of several variables by trigonometric polynomials whose harmonics lie in a hyperbolic cross and also properties of functions which do not have harmonics lying in a hyperbolic cross. Analogues of H. Bohr's inequality are obtained for such functions. Estimates of optimal order are obtained for the upper bounds of best approximations of certain classes of functions, defined using mixed differences, by trigonometric polynomials whose harmonics lie in a hyperbolic cross. The diameters of certain classes are found.Bibliography: 13 titles.