Symmetry

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27 pages, 503 KiB

Open AccessArticle

Holomorphic Extensions Associated with Fourier–Legendre Series and the Inverse Scattering Problem

by Enrico De Micheli

Symmetry 2021, 13(6), 1009; https://doi.org/10.3390/sym13061009 - 4 Jun 2021

Viewed by 1523

In this paper, we consider the inverse scattering problem and, in particular, the problem of reconstructing the spectral density associated with the Yukawian potentials from the sequence of the partial-waves

f_{ℓ}

of the Fourier–Legendre expansion of the scattering amplitude. We prove that [...] Read more.

In this paper, we consider the inverse scattering problem and, in particular, the problem of reconstructing the spectral density associated with the Yukawian potentials from the sequence of the partial-waves

f_{ℓ}

of the Fourier–Legendre expansion of the scattering amplitude. We prove that if the partial-waves

f_{ℓ}

satisfy a suitable Hausdorff-type condition, then they can be uniquely interpolated by a function

\tilde{f} (λ) \in C

, analytic in a half-plane. Assuming also the Martin condition to hold, we can prove that the Fourier–Legendre expansion of the scattering amplitude converges uniformly to a function

f (θ) \in C

(

θ

being the complexified scattering angle), which is analytic in a strip contained in the

θ

-plane. This result is obtained mainly through geometrical methods by replacing the analysis on the complex

cos θ

-plane with the analysis on a suitable complex hyperboloid. The double analytic symmetry of the scattering amplitude is therefore made manifest by its analyticity properties in the

λ

- and

θ

-planes. The function

f (θ)

is shown to have a holomorphic extension to a cut-domain, and from the discontinuity across the cuts we can iteratively reconstruct the spectral density

σ (μ)

associated with the class of Yukawian potentials. A reconstruction algorithm which makes use of Pollaczeck and Laguerre polynomials is finally given. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

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8 pages, 266 KiB

Open AccessArticle

Dense and σ-Porous Subsets in Some Families of Darboux Functions

by Gertruda Ivanova and Irena Domnik

Symmetry 2021, 13(5), 759; https://doi.org/10.3390/sym13050759 - 27 Apr 2021

Cited by 3 | Viewed by 1360

Abstract

G. Ivanova and E. Wagner-Bojakowska shown that the set of Darboux quasi-continuous functions with nowhere dense set of discontinuity points is dense in the metric space of Darboux quasi-continuous functions with the supremum metric. We prove that this set also is

σ

-strongly [...] Read more.

G. Ivanova and E. Wagner-Bojakowska shown that the set of Darboux quasi-continuous functions with nowhere dense set of discontinuity points is dense in the metric space of Darboux quasi-continuous functions with the supremum metric. We prove that this set also is

σ

-strongly porous in such space. We obtain the symmetrical result for the family of strong Świątkowski functions, i.e., that the family of strong Świątkowski functions with nowhere dense set of discontinuity points is dense (thus, “large”) and

σ

-strongly porous (thus, asymmetrically, “small”) in the family of strong Świątkowski functions. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

18 pages, 370 KiB

Open AccessArticle

Applications of Certain Conic Domains to a Subclass of q-Starlike Functions Associated with the Janowski Functions

by Bilal Khan, Hari Mohan Srivastava, Nazar Khan, Maslina Darus, Qazi Zahoor Ahmad and Muhammad Tahir

Symmetry 2021, 13(4), 574; https://doi.org/10.3390/sym13040574 - 31 Mar 2021

Cited by 25 | Viewed by 2788

Abstract

In our present investigation, with the help of the basic (or q-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of q-starlike functions, which maps the open unit [...] Read more.

In our present investigation, with the help of the basic (or q-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of q-starlike functions, which maps the open unit disk

U

, given by

U = \{z : z \in C and |z| < 1\},

onto this generalized conic type domain. We study here some such potentially useful results as, for example, the sufficient conditions, closure results, the Fekete-Szegö type inequalities and distortion theorems. We also obtain the lower bounds for the ratio of some functions which belong to this newly-defined function class and for the sequences of the partial sums. Our results are shown to be connected with several earlier works related to the field of our present investigation. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward

(p, q)

-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter p is obviously redundant. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

12 pages, 312 KiB

Open AccessArticle

Second Hankel Determinant for a Certain Subclass of Bi-Close to Convex Functions Defined by Kaplan

by Stanislawa Kanas, Pesse V. Sivasankari, Roy Karthiyayini and Srikandan Sivasubramanian

Symmetry 2021, 13(4), 567; https://doi.org/10.3390/sym13040567 - 29 Mar 2021

Cited by 4 | Viewed by 1849

Abstract

In this paper, we consider the class of strongly bi-close-to-convex functions of order

α

and bi-close-to-convex functions of order

β

. We obtain an upper bound estimate for the second Hankel determinant for functions belonging to these classes. The results in this article [...] Read more.

In this paper, we consider the class of strongly bi-close-to-convex functions of order

α

and bi-close-to-convex functions of order

β

. We obtain an upper bound estimate for the second Hankel determinant for functions belonging to these classes. The results in this article improve some earlier result obtained for the class of bi-convex functions. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

18 pages, 351 KiB

Open AccessArticle

Growth Analysis of Meromorphic Solutions of Linear Difference Equations with Entire or Meromorphic Coefficients of Finite φ-Order

by Junesang Choi, Sanjib Kumar Datta and Nityagopal Biswas

Symmetry 2021, 13(2), 267; https://doi.org/10.3390/sym13020267 - 5 Feb 2021

Cited by 2 | Viewed by 2180

Abstract

Many researchers’ attentions have been attracted to various growth properties of meromorphic solution f (of finite

φ

-order) of the following higher order linear difference equation [...] Read more.

Many researchers’ attentions have been attracted to various growth properties of meromorphic solution f (of finite

φ

-order) of the following higher order linear difference equation

A_{n} (z) f (z + n) + . . . + A_{1} (z) f (z + 1) + A_{0} (z) f (z) = 0

, where

A_{n} (z), \dots, A_{0} (z)

are entire or meromorphic coefficients (of finite

φ

-order) in the complex plane (

φ : [0, \infty) \to (0, \infty)

is a non-decreasing unbounded function). In this paper, by introducing a constant

b

(depending on

φ

) defined by

\underset{r \to \infty}{lim_{̲}} \frac{log r}{log φ (r)} = b < \infty,

and we show how nicely diverse known results for the meromorphic solution f of finite

φ

-order of the above difference equation can be modified. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

10 pages, 262 KiB

Open AccessEditor’s ChoiceArticle

New Conditions for Univalence of Confluent Hypergeometric Function

by Georgia Irina Oros

Symmetry 2021, 13(1), 82; https://doi.org/10.3390/sym13010082 - 5 Jan 2021

Cited by 16 | Viewed by 2547

Abstract

Since in many particular cases checking directly the conditions from the definitions of starlikeness or convexity of a function can be difficult, in this paper we use the theory of differential subordination and in particular the method of admissible functions in order to [...] Read more.

Since in many particular cases checking directly the conditions from the definitions of starlikeness or convexity of a function can be difficult, in this paper we use the theory of differential subordination and in particular the method of admissible functions in order to determine conditions of starlikeness and convexity for the confluent (Kummer) hypergeometric function of the first kind. Having in mind the results obtained by Miller and Mocanu in 1990 who used

a, c \in R

, for the confluent (Kummer) hypergeometric function, in this investigation a and c complex numbers are used and two criteria for univalence of the investigated function are stated. An example is also included in order to show the relevance of the original results of the paper. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

15 pages, 315 KiB

Open AccessArticle

Estimates of Coefficient Functionals for Functions Convex in the Imaginary-Axis Direction

by Paweł Zaprawa and Katarzyna Tra̧bka-Wiȩcław

Symmetry 2020, 12(10), 1736; https://doi.org/10.3390/sym12101736 - 20 Oct 2020

Cited by 1 | Viewed by 1730

Abstract

Let

C_{0} (h)

be a subclass of analytic and close-to-convex functions defined in the open unit disk by the formula

R e {(1 - z^{2}) f^{'} (z)} > 0

. In this [...] Read more.

Let

C_{0} (h)

be a subclass of analytic and close-to-convex functions defined in the open unit disk by the formula

R e {(1 - z^{2}) f^{'} (z)} > 0

. In this paper, some coefficient problems for

C_{0} (h)

are considered. Some properties and bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates of the difference and of sum of successive coefficients, bounds of the sum of the first n coefficients and bounds of the n-th coefficient. The obtained results are used to determine coefficient estimates for both functions convex in the imaginary-axis direction with real coefficients and typically real functions. Moreover, the sum of the first initial coefficients for functions with a positive real part and with a fixed second coefficient is estimated. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

10 pages, 239 KiB

Open AccessArticle

Some Results of Fekete-Szegö Type. Results for Some Holomorphic Functions of Several Complex Variables

by Renata Długosz and Piotr Liczberski

Symmetry 2020, 12(10), 1707; https://doi.org/10.3390/sym12101707 - 16 Oct 2020

Cited by 3 | Viewed by 1506

Abstract

This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real [...] Read more.

This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real part and which Temljakov transform

L f

has positive real part, respectively. The main result arise some sharp estimates of the Minkowski balance of a combination of 2-homogeneous and the square of 1-homogeneous polynomials occurred in power series expansion of functions from aforementioned families. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

6 pages, 241 KiB

Open AccessArticle

Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case

by Jan L. Cieśliński and Dzianis Zhalukevich

Symmetry 2020, 12(9), 1550; https://doi.org/10.3390/sym12091550 - 20 Sep 2020

Cited by 5 | Viewed by 2171

Abstract

Scators form a vector space endowed with a non-distributive product, in the hyperbolic case, have physical applications related to some deformations of special relativity (breaking the Lorentz symmetry) while the elliptic case leads to new examples of hypercomplex numbers and related notions of [...] Read more.

Scators form a vector space endowed with a non-distributive product, in the hyperbolic case, have physical applications related to some deformations of special relativity (breaking the Lorentz symmetry) while the elliptic case leads to new examples of hypercomplex numbers and related notions of holomorphicity. Until now, only a few particular cases of scator holomorphic functions have been found. In this paper we obtain all solutions of the generalized Cauchy–Riemann system which describes analogues of holomorphic functions in the

(1 + 2)

-dimensional scator space. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

7 pages, 712 KiB

Open AccessArticle

New Identities Dealing with Gauss Sums

by Wenpeng Zhang, Abdul Samad and Zhuoyu Chen

Symmetry 2020, 12(9), 1416; https://doi.org/10.3390/sym12091416 - 26 Aug 2020

Cited by 6 | Viewed by 2096

Abstract

In this article, we used the elementary methods and the properties of the classical Gauss sums to study the problem of calculating some Gauss sums. In particular, we obtain some interesting calculating formulas for the Gauss sums corresponding to the eight-order and twelve-order [...] Read more.

In this article, we used the elementary methods and the properties of the classical Gauss sums to study the problem of calculating some Gauss sums. In particular, we obtain some interesting calculating formulas for the Gauss sums corresponding to the eight-order and twelve-order characters modulo p, where p be an odd prime with

p = 8 k + 1

or

p = 12 k + 1

. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

8 pages, 275 KiB

Open AccessArticle

The Fifth Coefficient of Bazilevič Functions

by Oh Sang Kwon, Derek Keith Thomas and Young Jae Sim

Symmetry 2020, 12(8), 1228; https://doi.org/10.3390/sym12081228 - 26 Jul 2020

Viewed by 1913

Abstract

Let

f \in A

, the class of normalized analytic functions defined in the unit disk

D

, and be given by

f (z) = z + \sum_{n = 2}^{\infty} a_{n} z^{n}

for

z \in D

. [...] Read more.

Let

f \in A

, the class of normalized analytic functions defined in the unit disk

D

, and be given by

f (z) = z + \sum_{n = 2}^{\infty} a_{n} z^{n}

for

z \in D

. This paper presents a new approach to finding bounds for

| a_{n} |

. As an application, we find the sharp bound for

| a_{5} |

for the class

B_{1} (α)

of Bazilevič functions when

α \geq 1

. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

11 pages, 2936 KiB

Open AccessArticle

Subclasses of Starlike and Convex Functions Associated with the Limaçon Domain

by Vali Soltani Masih and Stanisława Kanas

Symmetry 2020, 12(6), 942; https://doi.org/10.3390/sym12060942 - 3 Jun 2020

Cited by 40 | Viewed by 4304

Abstract

Let

{ST}_{L} (s)

and

{CV}_{L} (s)

denote the family of analytic and normalized functions f in the unit disk

D : = \{z : | z | < 1\}

, such that the quantity [...] Read more.

Let

{ST}_{L} (s)

and

{CV}_{L} (s)

denote the family of analytic and normalized functions f in the unit disk

D : = \{z : | z | < 1\}

, such that the quantity

z f^{'} (z) / f (z)

or

1 + z f^{″} (z) / f^{'} (z)

respectively are lying in the region bounded by the limaçon

{[{(u - 1)}^{2} + v^{2} - s^{4}]}^{2} = 4 s^{2} [{(u - 1 + s^{2})}^{2} + v^{2}],

where

0 < s \leq 1 / \sqrt{2}

. The limaçon of Pascal is a curve that possesses properties which qualify it for the several applications in mathematics, statistics (hypothesis testing problem) but also in mechanics (fluid processing applications, known limaçon technology is employed to extract electrical power from low-grade heat, etc.). In this paper we present some results concerning the behavior of f on the classes

{ST}_{L} (s)

or

{CV}_{L} (s)

. Some appropriate examples are given. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

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