Derivatives Of Analytic Functions Research Articles

Carleson measures induced by higher order Schwarzian derivatives and derivatives of analytic functions

Let G be a finitely generated Fuchsian group of the second kind without any parabolic element and f be a univalent analytic function in the unit diskDcompatible withG. In this paper, westudy the higher order Schwarzian derivatives: ?n+1( f ) = ?? n( f )?(n?1) f?? f???n( f ), n ? 3, where ?3(f) stands for the Schwarzian derivatives of f, and Sn(f) = (f?) n?1/2 Dn( f ?)? n?1/2 , n ? 2. For p > 0, we show that if |?n(f)(z)|p(1?|z|2)p(n?1)?1dxdy (resp. |Sn( f )(z)|p(1 ? |z|2)pn?1dxdy) satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domainF ofG, then |?n( f )(z)|p(1?|z|2)p(n?1)?1dxdy (resp. |Sn( f )(z)|p(1?|z|2)pn?1dxdy) is a Carleson measure in D. Similarly, for p > 0 and a bounded analytic function f in the unit disk D compatible with G, we prove that if | f ?(z)|p(1 ? |z|2)p?1dxdy satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G, then |f?(z)|p(1 ? |z|2)p?1dxdy is a Carleson measure in D.

Estimates for analytic functions concerned with Hankel determinant

UDC 517.5 We give an upper bound of Hankel determinant of the first order for the classes of an analytic function. In addition, an evaluation with the Hankel determinant from below will be given for the second angular derivative of analytic function. For new inequalities, the results of Jack's lemma and Hankel determinant were used. Moreover, in a class of analytic functions on the unit disc, assuming the existence of an angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

On Hadamard Compositions of Gelfond–Leontiev Derivatives of Analytic Functions

On Hadamard Compositions of Gelfond–Leontiev Derivatives of Analytic Functions

On the Characteristic Properties for Subclasses of the Higher Derivative of Analytic Functions

In the present paper, we introduce two subclasses, S*C(,,g,s,d) and TS*C(, ,g, s,d), of analytic functions . Coefficients bounds for these subclasses are calculated.
 The main purpose of this article is to originate characteristic properties of the functions in the above subclasses.

Radial growth of the derivatives of analytic functions in Besov spaces

Abstract For 1 < p < ∞, the Besov space Bp consists of those functions f which are analytic in the unit disc 𝔻 = {z ∈ 𝔺 : |z| < 1} and satisfy ∫𝔻(1 − |z|2) p −2|f ′(z)| p dA(z) < ∞. The space B 2 reduces to the classical Dirichlet space 𝒟. It is known that if f ∈ 𝒟then |f ′(reiθ )| = o[(1 − r)−1/2], for almost every ∈ [0, 2π]. Hallenbeck and Samotij proved that this result is sharp in a very strong sense. We obtain substitutes of the above results valid for the spaces Bp (1 < p < ∞) an we give also an application of our them to questions concerning multipliers between Besov spaces.

Hadamard compositions of Gelfond–Leont’ev derivatives of analytic functions

Hadamard compositions of Gelfond–Leont’ev derivatives of analytic functions

Hadamard compositions of Gelfond-Leont'ev derivatives of analytic functions

For analytic functions f and g, the growth of the Hadamard composition of their Gelfond-Leont'ev derivatives is investigated in terms of generalized orders. A relation between the behaviors of the maximal terms of the Hadamard composition of Gelfond-Leont'ev derivatives and those of the Gelfond-Leont'ev derivative of a Hadamard composition is established.

Explicit Real-Part Estimates for High Order Derivatives of Analytic Functions

The representation for the sharp constant \(\mathrm{K}_{n, p}\) in an estimate of the modulus of the n-th derivative of an analytic function in the upper half-plane \({\mathbb C}_+\) is considered in this paper. It is assumed that the boundary value of the real part of the function on \(\partial {\mathbb C}_+\) belongs to \(L^p\). The representation for \(\mathrm{K}_{n, p}\) implies an optimization problem for a parameter in some integral. This problem is solved for \(p=2(m+1)/(2m+1-n)\), \(n\le 2m+1\), and for some first derivatives of even order in the case \(p=\infty \). The formula for \(\mathrm{K}_{n, 2(m+1)/(2m+1-n)}\) contains, for instance, the known expressions for \(\mathrm{K}_{2m+1, \infty }\) and \(\mathrm{K}_{m, 2}\) as particular cases. Also, a two-sided estimate for \(\mathrm{K}_{2m, \infty }\) is derived, which leads to the asymptotic formula \(\mathrm{K}_{2m, \infty }=2((2m-1))^2/\pi + O(((2m-1))^2/(2m-1))\) as \(m \rightarrow \infty \). The lower and upper bounds of \(\mathrm{K}_{2m, \infty }\) are compared with its value for the cases \(m=1, 2, 3, 4\). As applications, some real-part theorems with explicit constants for high order derivatives of analytic functions in subdomains of the complex plane are described.

Estimates for Growth of Derivatives of Analytic Functions Along the Radius

We study the radial boundary behavior of functions analytic in a unit disk of the complex plane. 1. The classical Hardy–Littlewood theorem [1] describes the relationship between the smoothness of boundary values of an analytic function on the boundary of a disk of analyticity and the rate of increase in the modulus of its higher-order derivatives. This theorem is an efficient tool for the solution of numerous problems of the theory of functions and the theory of trigonometric series. However, it is often necessary to estimate higher-order derivatives of an analytic function by using solely the information about the modulus of continuity of boundary values of the real part of this function. In the present paper, we consider the following problem: Let a function f be analytic in the disk D := {z ∈ C: |z| < 1} and let the function u := Re f be continuous in D. It is known that, for given n ∈ Z+, the function t �→u(e it ) can be represented in the form

A Real Part Theorem for the Higher Derivatives of Analytic Functions in the Unit Disk

Let \(n\) be a positive integer. Let \(\mathbf{U }\) be the unit disk, \(p\ge 1\) and let \(h^p(\mathbf{U })\) be the Hardy space of harmonic functions. Kresin and Maz’ya in a recent paper found a representation for the function \(H_{n,p}(z)\) in the inequality $$\begin{aligned}|f^{(n)} (z)|\le H_{n,p}(z)\Vert \mathfrak R (f-\mathcal{P }_{l})\Vert _{h^p(\mathbf{U })}, \quad \mathfrak R f\in h^p(\mathbf{U }), z\in \mathbf{U },\end{aligned}$$ where \(\mathcal{P }_{l}\) is a polynomial of degree \(l\le n-1\). We determine the sharp constant \(C_{p,n}\) in the inequality \(H_{n,p}(z)\le \frac{C_{p,n}}{(1-|z|^2)^{1/p+n}}\). This extends a recent result of Kalaj and Markovic, where only the case \(n=1\) was considered. As a corollary, an inequality for the modulus of \(n\)-th derivative of an analytic function defined in a complex domain with the bounded real part is obtained. This result improves a recent result of Kresin and Maz’ya.

Sharp and maximized real-part estimates for derivatives of analytic functions in the disk

Representations for the sharp coefficient in an estimate of the modulus of the $n$-th derivative of an analytic function in the unit disk ${\mathbb D}$ are obtained. It is assumed that the boundary value of the real part of the function on $\partial{\mathbb D}$ belongs to $L^p$. The maximum of a bounded factor in the representation of the sharp coefficient is found. Thereby, a pointwise estimate of the modulus of the $n$-th derivative of an analytic function in ${\mathbb D}$ with a best constant is obtained. The sharp coefficient in the estimate of the modulus of the first derivative in the explicit form is found. This coefficient is represented, for $p\in (1, \infty)$, as the product of monotonic functions of $|z|$.

Numerical evaluation of fractional integrals and derivatives of analytic function

A numerical technique based on the Cauchy integral formula of complex analysis has been employed for evaluating the integrals and derivatives of fractional orders of an analytic function. The method of subtraction of singularity has been used to evaluate the contour integrals for a desired degree of accuracy.

Wiman-Valiron Theory in Simply Connected Domains

We obtain asymptotic expressions for the derivatives of analytic functions in simply connected domains.

Generalization of theorems of Szász and Ruscheweyh on exact bounds for derivatives of analytic functions

Let Ω and Π be two domains in the extended complex plane equipped by the Poincare metric. In this paper we obtain analogs of Schwarz-Pick type inequalities in the class A(Ω, gH) = {f: Ω → Π} of functions locally holomorphic in Ω; for the domain Ω we consider the exterior of the unit disk and the upper half-plane. The obtained results generalize the well-known theorems of Szasz and Ruscheweyh about the exact estimates of derivatives of analytic functions defined on the disk |z| < 1.

Some integral formulas for analytic functions over noncommutative banach algebras

We establish a number of new formulas for analytic functions defined on certain special open subsets of a noncommutative complex Banach algebra. In particular, we find the analog of the Riesz-Dunford integral representation for the higher derivatives of analytic functions in noncommuting directions.

Schwarzian curvatures of holomorphic curves

In this paper, we define a generalization of the Schwarzian derivative of analytic functions to holomorphic curves in projective spaces, the Schwarzian curvatures. We study the curves with constant Schwarzian curvatures, the curves with prescribed Schwarzian curvatures, the Schwarzian curvatures of a given curve, and the transformation rules for the Schwarzian curvatures under a coordinate change.

Complex Interpolation with Derivatives of Analytic Functions

We develop the complex interpolation associated with derivatives of analytic functions such as the generalized Calderón′s inequality, norm estimates for duality, interpolation of operators, and so on. The present interpolation is available for regular interpolation families.

Complex interpolation functors with a family of quasi-power function parameters

For the complex interpolation functors associated with derivatives of analytic functions, the Calderon fundamental inequality is formulated in both additive and multiplicative forms; discretization, reiteration, the Calderon-Lozanovskii construction for B

Zeros of successive derivatives of analytic functions having a single essential singularity II

Zeros of successive derivatives of analytic functions having a single essential singularity II

Derivatives of analytic functions and bounded mean oscillation

Etude du comportement a la frontiere des fonctions analytiques dans le disque-unite. Caracterisation des fonctions a oscillation moyenne bornee