Littlewood–Paley conjecture associated with certain classes of analytic functions

Majorization properties for certain classes of analytic functions using the Sălăgean operator

In this paper, using Hadmard product, we introduce certain new classes of analytic functions in the open unit disk. Such results as inclusion and subordination relationships, characterization and coecient estimates, growth and distortion theorems, extreme points, closure theorems and radius of starlikeness and convexity belonging to the class TS(f;g;;;;A;B ) are obtained. Further subordination results for the class S(f;g;;;;A;B ) are derived:

We discuss the best linear approximation methods in the Hardy spaceH q q≥1, for classes of analytic functions studied by N. Ainulloev; these are generalizations (in a certain sense) of function sets introduced by L. V. Taikov. The exact values of their linear and Gelfandn-widths are obtained. The exact values of the Kolmogorov and Bernsteinn-widths of classes of analytic (in |z|<1) functions whose boundaryK-functionals are majorized by a prescribed functions are also obtained.

Let f (z )= D(F(z)) where D is a differential operator defined separately in every result. Let F be analytic and F(0 )= 0,F � (0 )= 1. We shall find out the disc in which operator D transforms some classes of analytic functions into the same. if and only if p1,p2 ∈ P(n,β) for z ∈ E. It is easy to see that P2(1,β) ≡ P(β) and P(1,0) ≡ P. The class Pk(1,0) ≡ Pk was introduced by Pinchuk (6), where he has generalized the concept of functions of bounded boundary rotation. It is worth mentioning that, for k > 2, functions in Pk need not be with the positive real part. It is easy to see that p ∈ Pk(n,β) if and only if there exists h ∈ Pk(n,0) such that p(z )=( 1 − β)h(z )+ β.

Let S l * denote the class of analytic functions f in the open unit disk D = { z : | z | &lt; 1 } normalized by f ( 0 ) = f ′ ( 0 ) − 1 = 0 , which is subordinate to exponential function, z f ′ ( z ) f ( z ) ≺ e z ( z ∈ D ) . In this paper, we aim to investigate the third-order Hankel determinant H 3 ( 1 ) for this function class S l * associated with exponential function and obtain the upper bound of the determinant H 3 ( 1 ) . Meanwhile, we give two examples to illustrate the results obtained.

This paper is devoted to the approximation of a certain class of analytic functions by shifts Z(s+iτ), τ∈R, of the modified Mellin transform Z(s) of the square of the Riemann zeta-function ζ(1/2+it). More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts Z(s+iτ), which approximate a given analytic function from F with a given accuracy. In the proof, the weak convergence of measures in the space of analytic functions is applied. Then, the set F coincides with the support of a limit measure.

In the present investigation, subclasses of analytic functions with respect to symmetrical points which are defined by the generalized Bessel functions of the first kind of orderμare introduced. Furthermore, some alluring geometric properties of these classes, which include inclusion property, integral-preserving properties, coefficients, and distortion results are studied. Moreover, some consequences of our results are also given.

On some applications of a subordination theorem

Radii of starlikeness and convexity for certain classes of analytic functions

By making use of the familiar concept of neighborhoods of analytic functions, we prove several inclusion relations associated with the (n, δ)neighborhoods of various subclass of univalent functions with negative coefficients.

This paper deals with the numerical continuation problem of analytic functions $g(z)$ given by $g(z) = \int_\Omega K(z,t) f(t)dt$ where K defines a bounded operator on $L^2 (\Omega )$. It is assumed that g is known over a finite segment A of the real line where g is to be sampled. Our continuation techniques emerged partly from a generalization of the Landweber iteration. We show that a certain discrete approximation of the proposed iterative technique yields two-step (noniterative) algorithms for solving the continuation problem. We also prove convergence of these approximations to the sought function g. Special emphasis on the continuation problem for the case $K(z,t) = e^{ - 2\pi izt} $ is given, and some related numerical examples are presented. The continuation problem when the known part of $g(z)$ is contaminated with some noise is addressed, and some techniques for solving this problem are also provided.

The objective of this paper is to present several results in theory of analytic functions (which are univalent, starlike- or prestarlike- and convex) by associating certain fractional calculus operators. A linear operator is fruitfully employed in our investigations, and applications to Wright's generalized hypergeometric functions are also discussed.

Radii of starlikeness and convexity in certain classes of analytic functions in the disk

Making use of a convolution structure, we introduce a new class of analytic functions PTg(λ,α,β,γ) defined in the open unit disc and investigate its various characteristics. Further we obtained distortion bounds , extreme points and radii of close-to-convexity, starlikeness and convexity for functions belonging to the class PTg(λ,α,β,γ).

In the present paper we introduce and studied two subclasses of multivalent functions denoted by M λ p,n(γ;β) and N λ p,n(μ,η;δ). Further, by giving specific values of the parameters of our main results, we will find some connection between these two classes, and moreover, several consequences of main results are also discussed.