scholarly journals Connecting quantum calculus and harmonic starlike functions

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1431-1441
Author(s):  
O.P. Ahuja ◽  
A. Çetinkaya
Keyword(s):  
Operator Theory ◽  
Harmonic Functions ◽  
Quantum Physics ◽  
Hypergeometric Series ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Quantum Calculus ◽  
Theory Approximation ◽  
Harmonic Starlike

Quantum calculus or q-calculus plays an important role in hypergeometric series, quantum physics, operator theory, approximation theory, sobolev spaces, geometric functions theory and others. But role of q-calculus in the theory of harmonic univalent functions is quite new. In this paper, we make an attempt to connect quantum calculus and harmonic univalent starlike functions. In particular, we introduce and investigate the properties of q-harmonic functions and q-harmonic starlike functions of order ?.

A Class of Harmonic Starlike Functions defined by Multiplier Transformation

2020 ◽  
pp. 455-469
Author(s):  
Deepali Khurana ◽  
Raj Kumar ◽  
Sibel Yalcin
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Harmonic Mappings ◽  
Distortion Bounds ◽  
Radius Of Convexity ◽  
Univalent Harmonic Mappings ◽  
Harmonic Starlike ◽  
Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

scholarly journals On Certain Subclass of Harmonic Starlike Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
A. Y. Lashin
Keyword(s):  
Integral Operator ◽  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disc ◽  
New Class ◽  
Distortion Bounds ◽  
Harmonic Starlike

Coefficient conditions, distortion bounds, extreme points, convolution, convex combinations, and neighborhoods for a new class of harmonic univalent functions in the open unit disc are investigated. Further, a class preserving integral operator and connections with various previously known results are briefly discussed.

scholarly journals Construction of Planar Harmonic Functions

2007 ◽  
Vol 2007 ◽  
pp. 1-11
Author(s):  
Jay M. Jahangiri ◽  
Herb Silverman ◽  
Evelyn M. Silvia
Keyword(s):  
Unit Disk ◽  
Harmonic Functions ◽  
Harmonic Maps ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Harmonic Starlike ◽  
Complex Valued

Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk can be written in the formf=h+g¯, wherehandgare analytic in the open unit disk. The functionshandgare called the analytic and coanalytic parts off, respectively. In this paper, we construct certain planar harmonic maps either by varying the coanalytic parts of harmonic functions that are known to be harmonic starlike or by adjoining analytic univalent functions with coanalytic parts that are related or derived from the analytic parts.

scholarly journals Applications of the q-Srivastava-Attiya Operator Involving a Certain Family of Bi-Univalent Functions Associated with the Horadam Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1230
Author(s):  
Hari Mohan Srivastava ◽  
Abbas Kareem Wanas ◽  
Rekha Srivastava
Keyword(s):  
Univalent Functions ◽  
Zeta Functions ◽  
Starlike Functions ◽  
Function Class ◽  
Open Unit ◽  
Quantum Calculus ◽  
New Family ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Local Symmetries

In this article, by making use of the q-Srivastava-Attiya operator, we introduce and investigate a new family SWΣ(δ,γ,λ,s,t,q,r) of normalized holomorphic and bi-univalent functions in the open unit disk U, which are associated with the Bazilevič functions and the λ-pseudo-starlike functions as well as the Horadam polynomials. We estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to the holomorphic and bi-univalent function class, which we introduce here. Furthermore, we establish the Fekete-Szegö inequality for functions in the family SWΣ(δ,γ,λ,s,t,q,r). Relevant connections of some of the special cases of the main results with those in several earlier works are also pointed out. Our usage here of the basic or quantum (or q-) extension of the familiar Hurwitz-Lerch zeta function Φ(z,s,a) is justified by the fact that several members of this family of zeta functions possess properties with local or non-local symmetries. Our study of the applications of such quantum (or q-) extensions in this paper is also motivated by the symmetric nature of quantum calculus itself.

scholarly journals Applications of $ q $-difference symmetric operator in harmonic univalent functions

2021 ◽  
Vol 7 (1) ◽  
pp. 667-680
Author(s):  
Caihuan Zhang ◽  
◽  
Shahid Khan ◽  
Aftab Hussain ◽  
Nazar Khan ◽  
...  
Keyword(s):  
Differential Operator ◽  
Operator Theory ◽  
Harmonic Functions ◽  
Symmetric Operator ◽  
Univalent Functions ◽  
Coefficient Bounds ◽  
New Class ◽  
Coefficient Condition ◽  
Distortion Theorems ◽  
First Time

<abstract><p>In this paper, for the first time, we apply symmetric $ q $ -calculus operator theory to define symmetric Salagean $ q $-differential operator. We introduce a new class $ \widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) $ of harmonic univalent functions $ f $ associated with newly defined symmetric Salagean $ q $-differential operator for complex harmonic functions. A sufficient coefficient condition for the functions $ f $ to be sense preserving and univalent and in the same class is obtained. It is proved that this coefficient condition is necessary for the functions in its subclass $ \overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) } $ and obtain sharp coefficient bounds, distortion theorems and covering results. Furthermore, we also highlight some known consequence of our main results.</p></abstract>

It Keeps Me Seeking

2018 ◽  
Author(s):  
Andrew Briggs ◽  
Hans Halvorson ◽  
Andrew Steane
Keyword(s):  
Evolutionary Biology ◽  
Quantum Physics ◽  
Natural World ◽  
Theological Reflection ◽  
Fine Tuning ◽  
Good Argument ◽  
The Difference ◽  
Argument From Design ◽  
Relationship Of

Two scientists and a philosopher aim to show how science both enriches and is enriched by Christian faith. The text is written around four themes: 1. God is a being to be known, not a hypothesis to be tested; 2. We set a high bar on what constitutes good argument; 3. Uncertainty is OK; 4. We are allowed to open up the window that the natural world offers us. This is not a work of apologetics. Rather, the text takes an overview of various themes and gives reactions and responses, intended to place science correctly as a valued component of the life of faith. The difference between philosophical analysis and theological reflection is expounded. Questions of human identity are addressed from philosophy, computer science, quantum physics, evolutionary biology and theological reflection. Contemporary physics reveals the subtle and open nature of physical existence, and offers lessons in how to learn and how to live with incomplete knowledge. The nature and role of miracles is considered. The ‘argument from design’ is critiqued, especially arguments from fine-tuning. Logical derivation from impersonal facts is not an appropriate route to a relationship of mutual trust. Mainstream evolutionary biology is assessed to be a valuable component of our understanding, but no exploratory process can itself fully account for the nature of what is discovered. To engage deeply in science is to seek truth and to seek a better future; it is also an activity of appreciation, as one may appreciate a work of art.

scholarly journals General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1102
Author(s):  
Yashoverdhan Vyas ◽  
Hari M. Srivastava ◽  
Shivani Pathak ◽  
Kalpana Fatawat
Keyword(s):  
Hypergeometric Functions ◽  
Theory Theory ◽  
Binomial Theorem ◽  
Generalized Hypergeometric Functions ◽  
Quantum Calculus ◽  
The Third ◽  
Summation Formulas ◽  
Symmetric Quantum ◽  
Summation Theorem

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

scholarly journals The Invariant Set Postulate: a new geometric framework for the foundations of quantum theory and the role played by gravity

2009 ◽  
Vol 465 (2110) ◽  
pp. 3165-3185 ◽  
Author(s):  
T. N. Palmer
Keyword(s):  
Quantum Theory ◽  
Science Fiction ◽  
State Space ◽  
Fractal Geometry ◽  
Quantum Physics ◽  
Nonlinear Dynamical Systems ◽  
Physical Reality ◽  
Invariant Set ◽  
Geometric Framework

A new law of physics is proposed, defined on the cosmological scale but with significant implications for the microscale. Motivated by nonlinear dynamical systems theory and black-hole thermodynamics, the Invariant Set Postulate proposes that cosmological states of physical reality belong to a non-computable fractal state-space geometry I , invariant under the action of some subordinate deterministic causal dynamics D I . An exploratory analysis is made of a possible causal realistic framework for quantum physics based on key properties of I . For example, sparseness is used to relate generic counterfactual states to points p ∉ I of unreality, thus providing a geometric basis for the essential contextuality of quantum physics and the role of the abstract Hilbert Space in quantum theory. Also, self-similarity, described in a symbolic setting, provides a possible realistic perspective on the essential role of complex numbers and quaternions in quantum theory. A new interpretation is given to the standard ‘mysteries’ of quantum theory: superposition, measurement, non-locality, emergence of classicality and so on. It is proposed that heterogeneities in the fractal geometry of I are manifestations of the phenomenon of gravity. Since quantum theory is inherently blind to the existence of such state-space geometries, the analysis here suggests that attempts to formulate unified theories of physics within a conventional quantum-theoretic framework are misguided, and that a successful quantum theory of gravity should unify the causal non-Euclidean geometry of space–time with the atemporal fractal geometry of state space. The task is not to make sense of the quantum axioms by heaping more structure, more definitions, more science fiction imagery on top of them, but to throw them away wholesale and start afresh. We should be relentless in asking ourselves: From what deep physical principles might we derive this exquisite structure? These principles should be crisp, they should be compelling. They should stir the soul. Chris Fuchs ( Gilder 2008 , p. 335)

scholarly journals Harmonic univalent functions defined by post quantum calculus operators

2019 ◽  
Vol 11 (1) ◽  
pp. 5-17 ◽  
Author(s):  
Om P. Ahuja ◽  
Asena Çetinkaya ◽  
V. Ravichandran
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Quantum Calculus ◽  
Special Cases ◽  
The Family ◽  
Covering Theorems

Abstract We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

scholarly journals Coefficient Estimates and the Fekete–Szegö Problem for New Classes of m-Fold Symmetric Bi-Univalent Functions

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 129
Author(s):  
Georgia Irina Oros ◽  
Luminiţa-Ioana Cotîrlă
Keyword(s):  
Univalent Functions ◽  
Coefficient Estimates ◽  
Quantum Calculus

The results presented in this paper deal with the classical but still prevalent problem of introducing new classes of m-fold symmetric bi-univalent functions and studying properties related to coefficient estimates. Quantum calculus aspects are also considered in this study in order to enhance its novelty and to obtain more interesting results. We present three new classes of bi-univalent functions, generalizing certain previously studied classes. The relation between the known results and the new ones presented here is highlighted. Estimates on the Taylor–Maclaurin coefficients |am+1| and |a2m+1| are obtained and, furthermore, the much investigated aspect of Fekete–Szegő functional is also considered for each of the new classes.

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