Research

14 pages, 444 KiB

Open AccessArticle

Coefficient Inequalities for q-Convex Functions with Respect to q-Analogue of the Exponential Function

by Majid Khan, Nazar Khan, Ferdous M. O. Tawfiq and Jong-Suk Ro

Axioms 2023, 12(12), 1130; https://doi.org/10.3390/axioms12121130 - 15 Dec 2023

Viewed by 885

In mathematical analysis, the q-analogue of a function refers to a modified version of the function that is derived from q-series expansions. This paper is focused on the q-analogue of the exponential function and investigates a class of convex functions [...] Read more.

In mathematical analysis, the q-analogue of a function refers to a modified version of the function that is derived from q-series expansions. This paper is focused on the q-analogue of the exponential function and investigates a class of convex functions associated with it. The main objective is to derive precise inequalities that bound the coefficients of these convex functions. In this research, the initial coefficient bounds, Fekete–Szegő problem, second and third Hankel determinant have been determined. These coefficient bounds provide valuable information about the behavior and properties of the functions within the considered class. Full article

(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))

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14 pages, 370 KiB

Open AccessArticle

Plurisubharmonic Interpolation and Plurisubharmonic Geodesics

by Alexander Rashkovskii

Axioms 2023, 12(7), 671; https://doi.org/10.3390/axioms12070671 - 07 Jul 2023

Viewed by 794

Abstract

We give a short survey on plurisubharmonic interpolation, with a focus on the possibility of connecting two given plurisubharmonic functions by plurisubharmonic geodesics. Full article

(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))

11 pages, 1384 KiB

Open AccessArticle

Radius of Uniformly Convex γ-Spirallikeness of Combination of Derivatives of Bessel Functions

by Stanislawa Kanas and Kamaljeet Gangania

Axioms 2023, 12(5), 468; https://doi.org/10.3390/axioms12050468 - 12 May 2023

Viewed by 778

Abstract

We find the sharp radius of uniformly convex

γ

-spirallikeness for

N_{ν} (z) = a z^{2} J_{ν}^{″} (z) + b z J_{ν}^{'} (z) + c J_{ν} (z)

(here [...] Read more.

We find the sharp radius of uniformly convex

γ

-spirallikeness for

N_{ν} (z) = a z^{2} J_{ν}^{″} (z) + b z J_{ν}^{'} (z) + c J_{ν} (z)

(here

J_{ν} (z)

is the Bessel function of the first kind of order

ν

) with three different kinds of normalizations of the function

N_{ν} (z)

. As an application, we derive sufficient conditions on the parameters for the functions to be uniformly convex

γ

-spirallikeness and, consequently, generate examples of uniform convex

γ

-spirallike via

N_{ν} (z)

. Results are well-supported by the relevant graphs and tables. Full article

(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))

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13 pages, 304 KiB

Open AccessArticle

Logarithmic Coefficients for Some Classes Defined by Subordination

by Ebrahim Analouei Adegani, Ahmad Motamednezhad, Teodor Bulboacă and Nak Eun Cho

Axioms 2023, 12(4), 332; https://doi.org/10.3390/axioms12040332 - 29 Mar 2023

Cited by 2 | Viewed by 727

Abstract

In this paper, we obtain the sharp and accurate bounds for the logarithmic coefficients of some subclasses of analytic functions defined and studied in earlier works. Furthermore, we obtain the bounds of the second Hankel determinant of logarithmic coefficients for a class defined [...] Read more.

In this paper, we obtain the sharp and accurate bounds for the logarithmic coefficients of some subclasses of analytic functions defined and studied in earlier works. Furthermore, we obtain the bounds of the second Hankel determinant of logarithmic coefficients for a class defined by subordination, such as the class of starlike functions

S^{*} (φ)

. Some applications of our results, which are extensions of those reported in earlier papers are given here as special cases. In addition, the results given can be used for other popular subclasses. Full article

(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))

14 pages, 327 KiB

Open AccessArticle

Entire Gaussian Functions: Probability of Zeros Absence

by Andriy Kuryliak and Oleh Skaskiv

Axioms 2023, 12(3), 255; https://doi.org/10.3390/axioms12030255 - 01 Mar 2023

Cited by 2 | Viewed by 814

Abstract

In this paper, we consider a random entire function of the form

f (z, ω) = \sum_{n = 0}^{+ \infty} ε_{n} (ω_{1})

\times ξ_{n} (ω_{2}) f_{n} z^{n},

[...] Read more.

In this paper, we consider a random entire function of the form

f (z, ω) = \sum_{n = 0}^{+ \infty} ε_{n} (ω_{1})

\times ξ_{n} (ω_{2}) f_{n} z^{n},

where

(ε_{n})

is a sequence of independent Steinhaus random variables,

(ξ_{n})

is the a sequence of independent standard complex Gaussian random variables, and a sequence of numbers

f_{n} \in C

is such that

\underset{n \to + \infty}{lim^{¯}} \sqrt[n]{| f_{n} |} = 0

and

# {n : f_{n} \neq 0} = + \infty .

We investigate asymptotic estimates of the probability

P_{0} (r) = P {ω : f (z, ω)

has no zeros inside

r D}

as

r \to + \infty

outside of some set E of finite logarithmic measure, i.e.,

\int_{E \cap [1, + \infty)} d \ln r < + \infty

. The obtained asymptotic estimates for the probability of the absence of zeros for entire Gaussian functions are in a certain sense the best possible result. Furthermore, we give an answer to an open question of A. Nishry for such random functions. Full article

(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))

16 pages, 837 KiB

Open AccessArticle

Deterministic and Random Generalized Complex Numbers Related to a Class of Positively Homogeneous Functionals

by Wolf-Dieter Richter

Axioms 2023, 12(1), 60; https://doi.org/10.3390/axioms12010060 - 04 Jan 2023

Cited by 2 | Viewed by 1079

Abstract

Based upon a new general vector-valued vector product, generalized complex numbers with respect to certain positively homogeneous functionals including norms and antinorms are introduced and a vector-valued Euler type formula for them is derived using a vector valued exponential function. Furthermore, generalized Cauchy–Riemann [...] Read more.

Based upon a new general vector-valued vector product, generalized complex numbers with respect to certain positively homogeneous functionals including norms and antinorms are introduced and a vector-valued Euler type formula for them is derived using a vector valued exponential function. Furthermore, generalized Cauchy–Riemann differential equations for generalized complex differentiable functions are derived. For random versions of the considered new type of generalized complex numbers, moments are introduced and uniform distributions on discs with respect to functionals of the considered type are analyzed. Moreover, generalized uniform distributions on corresponding circles are studied and a connection with generalized circle numbers, which are natural relatives of

π

, is established. Finally, random generalized complex numbers are considered which are star-shaped distributed. Full article

(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))

17 pages, 329 KiB

Open AccessArticle

Certain Geometric Properties of the Fox–Wright Functions

by Anish Kumar, Saiful R. Mondal and Sourav Das

Axioms 2022, 11(11), 629; https://doi.org/10.3390/axioms11110629 - 09 Nov 2022

Cited by 2 | Viewed by 1168

Abstract

The primary objective of this study is to establish necessary conditions so that the normalized Fox–Wright functions possess certain geometric properties, such as convexity and pre-starlikeness. In addition, we present a linear operator associated with the Fox–Wright functions and discuss its k-uniform [...] Read more.

The primary objective of this study is to establish necessary conditions so that the normalized Fox–Wright functions possess certain geometric properties, such as convexity and pre-starlikeness. In addition, we present a linear operator associated with the Fox–Wright functions and discuss its k-uniform convexity and k-uniform starlikeness. Furthermore, some sufficient conditions were obtained so that this function belongs to the Hardy spaces. The results of this work are presumably new and illustrated by several consequences, remarks, and examples. Full article

(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))

12 pages, 292 KiB

Open AccessArticle

On Holomorphic Contractibility of Teichmüller Spaces

by Samuel L. Krushkal

Axioms 2022, 11(10), 548; https://doi.org/10.3390/axioms11100548 - 12 Oct 2022

Viewed by 840

Abstract

The problem of the holomorphic contractibility of Teichmüller spaces

T (0, n)

of the punctured spheres (

n > 4

) arose in the 1970s in connection with solving algebraic equations in Banach algebras. Recently it was solved by the [...] Read more.

The problem of the holomorphic contractibility of Teichmüller spaces

T (0, n)

of the punctured spheres (

n > 4

) arose in the 1970s in connection with solving algebraic equations in Banach algebras. Recently it was solved by the author. In the present paper, we give a refined proof of the holomorphic contractibility for all spaces

T (0, n), n > 4

and provide two independent proofs of holomorphic contractibility for low-dimensional Teichmüller spaces, which has intrinsic interest. Full article

(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))

14 pages, 295 KiB

Open AccessArticle

Hadamard Compositions of Gelfond–Leont’ev Derivatives

by Myroslav Sheremeta

Axioms 2022, 11(9), 478; https://doi.org/10.3390/axioms11090478 - 18 Sep 2022

Cited by 1 | Viewed by 982

Abstract

For analytic functions

f_{j} (z) = \sum_{n = 0}^{\infty} a_{n, j} z^{n}

,

1 \leq j \leq p

, the notion of a Hadamard composition [...] Read more.

For analytic functions

f_{j} (z) = \sum_{n = 0}^{\infty} a_{n, j} z^{n}

,

1 \leq j \leq p

, the notion of a Hadamard composition

{(f_{1} * \dots * f_{p})}_{m} = \sum_{n = 0}^{\infty} (\sum_{k_{1} + \dots + k_{p} = m} c_{k_{1} \dots k_{p}} a_{n, 1}^{k_{1}} \cdot \dots \cdot a_{n, p}^{k_{p}}) z^{n}

of genus m is introduced. The relationship between the growth of the Gelfond–Leont’ev derivative of the Hadamard composition of functions

f_{j}

and the growth Hadamard composition of Gelfond–Leont’ev derivatives of these functions is studied. We found conditions under which these derivatives and the composition have the same order and a lower order. For the maximal terms of the power expansion of these derivatives, I describe behavior of their ratios. Full article

(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))

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