On the sum of squares of the coefficients of Bloch functions

In this article several types of inequalities for weighted sums of the moduli of Taylor coefficients for Bloch functions are provedComment: 13 page

Makarov's principle for the Bloch unit ball

Makarov's principle relates three characteristics of Bloch functions that resemble the variance of a Gaussian: asymptotic variance, the constant in Makarov's law of iterated logarithm and the second derivative of the integral means spectrum at the origin. While these quantities need not be equal in general, we show that the universal bounds agree if we take the supremum over the Bloch unit ball. For the supremum (of either of these quantities), we give the estimate Σ^2_B < min(0.9, Σ^2), where Σ^2 is the analogous quantity associated to the unit ball in the L∞ norm on the Bloch space. This improves on the upper bound in Pommerenke's estimate 0.685^2 < Σ^2_B ⩽ 1

Integral bounds for simple partial fractions

For p ≥ 2 we obtain bounds for L p-norms of the Fourier transform of real parts of simple partial fractions. For even p our estimate is sharp. We also prove a new inequality for L p-norms of simple partial fractions which in some cases is stronger than the corresponding inequality obtained by V. Yu. Protasov. © Allerton Press, Inc., 2012

Lower estimate for the integral means spectrum for p = -1

In this paper we show that there exists a function f bounded and univalent in the unit disk, such that ∫ |f′(reiθ)|-1dθ ≥ C(1 - r)-0.127, 0 ≤ r ≤ 1

Boundary behavior of the power series of the enlarged Bloch class

We construct the classes of functions analytic in the unit disk, other than lacunary, and possessing the following property: If a function belongs to one of these classes and the sum of the squares of its coefficients is unbounded then there exists a set of positive measure on the circle where the function has no radial limits. © 2009 Pleiades Publishing, Ltd

Sharp estimates for integral means for three classes of domains

In this paper, the following sharp estimate is proved: ∫ 0 2π|F′(e iθ)| p, dθ ≤ √π2 1+pΓ(1/2+p/2)/Γ(1+p/2)}, p>-1, where F is the conformal mapping of the domain D - = {ζ:|ζ| > 1} onto the exterior of a convex curve, with F'(\infty)=\nomathbreak 1. For p=\nomathbreak 1, this result is due to Pólya and Shiffer. We also obtain several generalizations of this estimate under other geometric assumptions about the structure of the domain F(D -)

A note on an area-type functional of Bloch functions

© 2017, Pleiades Publishing, Ltd.Let D be the unit disk centered at the origin in the complex plane. In this paper we consider an extremal problem for an area-type functional in the space B of Bloch functions with the seminorm ||b||B = sup{(1 − |z|2)|b’(z)|: z ∈ D}. We show that sup{Σk=1n k|bk|2: ||b||B ≤ 1} = nBn2, n = 1, 2, 3, 4, 5, where bk are the Taylor coefficients of b and Bn = sup{|bn|: ||b||B ≤ 1}

On Brennan's conjecture for a special class of functions

In this paper, we prove Brennan's conjecture for conformal mappings f of the disk {z : | z| < 1} assuming that the Taylor coefficients of the function log(zf′(z)/f(z)) at zero are nonnegative. We also obtain inequalities for the integral means over the circle |z| = r of the squared modulus of the function zf′(z)/f(z). © 2005 Springer Science+Business Media, Inc

Design characteristics of road embankments made of sandy soils

Sandy soils are widespread all over the world, including in Uzbekistan, and they are widely used in the construction of highways. Therefore, several regulatory documents have been developed that normalize their design characteristics. However, in the existing regulatory documents, the design characteristics of sandy soils of road embankments of highways used in the design of road coverings (modulus of elasticity E, angle of internal friction φ, specific adhesion C) are not normalized depending on the degree of compaction, calculated humidity and the content of dusty and clay particles in such soils. To solve these problems, special laboratory and field studies were conducted, the results of which are given in this article