Third-Order Hankel Determinant for Certain Class of Analytic Functions Related with Exponential Function

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:

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

In Geometric Function Theory, it is well known that the familiar Koebe function f(z) = z/(1 − z)2 is the extremal function for the class 𝒮* of starlike functions in the open unit disk 𝕌 and also that the function g(z) = z/(1 − z) is the extremal function for the class 𝒦 of convex functions in the open unit disk 𝕌. However, the partial sum fn (z) of f(z) is not starlike in 𝕌 and the partial sum gn (z) of g(z) is not convex in 𝕌. The aim of the present paper is to investigate the starlikeness and convexity of these partial sums fn (z) and gn (z). Computational and graphical usages of Mathematica (Version 4.0) as well as geometrical descriptions of the image domains in several illustrative examples are also presented. E-mail: owa@math.kindai.ac.jp

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.

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 this article, the present study employs the utilization of the concepts pertaining to (x,y)-symmetrical functions, Janowski type functions, and q-calculus in order to establish a novel subclass within the open unit disk. Specifically, we delve into the examination of convolution properties, which serve as a tool for investigating and inferring adequate and equivalent conditions. Moreover, we also explore specific characteristics of the class S˜qx,y(α,β,λ), thereby further scrutinizing the convolution properties of these newly defined classes.

A characterization theorem for a certain class of analytic functions

It is well known that there are two important classes of analytic functions of Ma-Minda type (MMT): Ma-Minda starlike and Ma-Minda convex functions. In this work, we suggest a new class of analytic functions, which is normalized in the open unit disk. The suggested class is generated by a roulette curve formula, which satisfies the symmetric behavior in the open unit disk. A roulette curve is shaped as the path outlined by the sum of two complex numbers, each affecting at a uniform rapidity in a circle. Special cases are illustrated involving special functions. Graphics of the curve are illustrated by using Mathematica 13.3.

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.

In this paper, new Ma–Minda type subclasses of analytic functions of complex order γ, associated with cardioid domain in the open unit disk , are introduced. The objective of this work is to investigate the first three sharp coefficient bounds and to examine the sharp estimates of the Fekete-Szegö problem for these subclasses.

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.

Let <TEX>$\mathcal{A}$</TEX> be the class of analytic functions f in the open unit disk <TEX>$\mathbb{U}$</TEX>={z : <TEX>${\mid}z{\mid}$</TEX> < 1} with the normalization conditions <TEX>$f(0)=f^{\prime}(0)-1=0$</TEX>. If <TEX>$f(z)=z+\sum_{n=2}^{\infty}a_nz^n$</TEX> and <TEX>${\delta}$</TEX> > 0 are given, then the <TEX>$T_{\delta}$</TEX>-neighborhood of the function f is defined as <TEX>$$TN_{\delta}(f)\{g(z)=z+\sum_{n=2}^{\infty}b_nz^n{\in}\mathcal{A}:\sum_{n=2}^{\infty}T_n{\mid}a_n-b_n{\mid}{\leq}{\delta}\}$$</TEX>, where <TEX>$T=\{T_n\}_{n=2}^{\infty}$</TEX> is a sequence of positive numbers. In the present paper we investigate some problems concerning <TEX>$T_{\delta}$</TEX>-neighborhoods of function in various classes of analytic functions with <TEX>$T=\{2^{-n}/n^2\}_{n=2}^{\infty}$</TEX>. We also find bounds for <TEX>$^{\delta}^*_T(A,B)$</TEX> defined by <TEX>$$^{\delta}^*_T(A,B)=jnf\{{\delta}</TEX><TEX>&</TEX><TEX>gt;0:B{\subset}TN_{\delta}(f)\;for\;all\;f{\in}A\}$$</TEX> where A, B are given subsets of <TEX>$\mathcal{A}$</TEX>.

"Strongly Janowski functions are used to define certain classes of analytic functions which generalize the concepts of close-to-convexity and bounded boundary rotation. Coefficient results, a necessary condition, distortion bounds, Hankel determinant problem and several other interesting properties of these classes are studied. Some significant well-known results are derived as special cases. Keywords: Starlike, convex, bounded boundary rotation, Carathéodory and Janowski functions, subordination, convolution."

Littlewood–Paley conjecture associated with certain classes of analytic functions

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.