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An analysis of the harmonic series reveals a geometric structure of atomic hydrogen that defines the minimum radius of the bound electron. Subsequently, hydrogen's energy states, velocities, and angular momentum are determined. Based on these findings, a theoretical value is determined for Planck's reduced constant and the electron rest mass. In addition, an examination of hydrogen's quantum energy states suggests that they are the result of a combinatorial effect. [ABSTRACT FROM AUTHOR]

This article focuses on the improvement of the classic Bohr’s inequality for bounded analytic functions on the unit disk. We give some sharp versions of Bohr’s inequality, generalizing the previous results. [ABSTRACT FROM AUTHOR]

Let H be the class of harmonic functions f = h + g ¯ in the unit disk D : = { z ∈ C : | z | < 1 } , where h and g are analytic in D . In 2020, N. Ghosh and V. Allu introduced the class P H 0 (M) of normalized harmonic mappings defined by P H 0 (M) = { f = h + g ¯ ∈ H : Re (z h ′ ′ (z)) > - M + | z g ′ ′ (z) | with M > 0 , g ′ (0) = 0 , z ∈ D } . In this paper, we investigate various geometric properties such as starlikeness, convexity, convex combination and convolution for functions in the class P H 0 (M) . Furthermore, we determine the sharp Bohr–Rogosinski radius, improved Bohr radius and refined Bohr radius for the class P H 0 (M) . [ABSTRACT FROM AUTHOR]

We study a "p-powered" version K n p ⁢ (F ⁢ (R)) of the well-known Bohr radius problem for the family F ⁢ (R) of holomorphic functions f : R → X satisfying ∥ f ∥ < ∞ , where ∥ ⋅ ∥ is a norm in the function space F ⁢ (R) , R ⊂ ℂ n is a complete Reinhardt domain, and X is a complex Banach space. For all p > 0 , we describe in full detail the asymptotic behavior of K n p ⁢ (F ⁢ (R)) , where F ⁢ (R) is: (a) the Hardy space of X-valued holomorphic functions defined in the open unit polydisk 픻 n ; and (b) the space of bounded X-valued holomorphic or complex-valued pluriharmonic functions defined in the open unit ball B ⁢ (l t n) of the Minkowski space l t n . We give an alternative definition of the optimal cotype for a complex Banach space X in the light of these results. In addition, the best possible versions of two theorems from [C. Bénéteau, A. Dahlner and D. Khavinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 2004, 1, 1–19] and [S. Chen and H. Hamada, Some sharp Schwarz–Pick type estimates and their applications of harmonic and pluriharmonic functions, J. Funct. Anal. 282 2022, 1, Paper No. 109254] have been obtained as specific instances of our results. [ABSTRACT FROM AUTHOR]

A class $ \mathcal {F} $ F consisting of analytic functions $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ f (z) = ∑ n = 0 ∞ a n z n in the unit disc $ \mathbb {D}=\{z\in \mathbb {C}:|z| D = { z ∈ C : | z | < 1 } satisfies a Bohr phenomenon if there exists an $ r_f>0 $ r f > 0 such that \[ \sum_{n=1}^{\infty}|a_n|r^n\leq{d}\left(f(0),\partial f(\mathbb{D})\right) \] ∑ n = 1 ∞ | a n | r n ≤ d (f (0) , ∂f (D)) for every function $ f\in \mathcal {F} $ f ∈ F , and $ |z|=r\leq r_f $ | z | = r ≤ r f . The largest radius $ r_f $ r f is the Bohr radius and the inequality $ \sum _{n=1}^{\infty }|a_n|r^n\leq {d}\left (f(0),\partial f(\mathbb {D})\right) $ ∑ n = 1 ∞ | a n | r n ≤ d (f (0) , ∂f (D)) is Bohr inequality for the class $ \mathcal {F} $ F , where 'd' is the Euclidean distance. In this paper, we prove sharp refinement of the Bohr–Rogosinski inequality for certain classes of harmonic mappings. [ABSTRACT FROM AUTHOR]

Let H 1 be the Euclidean space C m or ℓ 2 and let B H 1 be the unit ball of H 1 . In this paper, we will give new generalizations of several results related to the Bohr radius for locally univalent harmonic functions on the unit disc U in C to pluriharmonic mappings on B H 1 with values in H 1 which satisfy some assumptions. All of the results are sharp, and some of the results give improvements of the results on the unit disc. [ABSTRACT FROM AUTHOR]

The focus of this article is to explore the Bohr inequality for a specific subset of harmonic starlike mappings introduced by Ghosh and Vasudevarao (Some basic properties of certain subclass of harmonic univalent functions, Complex Var. Elliptic Equ. 63 (2018), no. 12, 1687–1703.). This set is denoted as ℬH0(M)≔{f=h+g¯∈ℋ0:∣zh″(z)∣≤M−∣zg″(z)∣}{{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M):= \{f=h+\overline{g}\in {{\mathcal{ {\mathcal H} }}}_{0}:| z{h}^{^{\prime\prime} }\left(z)| \le M-| z{g}^{^{\prime\prime} }\left(z)| \} for z∈Dz\in {\mathbb{D}}, where 0

Let $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain Ω containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z| \lt 1\}$ into a complex Banach space X with $\left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1$. Let $\phi=\{\phi_{n}(r)\}_{n=0}^{\infty}$ with $\phi_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} \phi_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q \lt \infty$ , we denote \begin{equation*} R_{p,q,\phi}(f,\Omega,X)= \sup \left\{r \geq 0: \left\lVert x_{0}\right\rVert^p \phi_{0}(r) + \left(\sum_{n=1}^{\infty} \left\lVert x_{n}\right\rVert\phi_{n}(r)\right)^q \leq \phi_{0}(r)\right\} \end{equation*} and define the Bohr radius associated with ϕ by \begin{equation*}R_{p,q,\phi}(\Omega,X)=\inf \left\{R_{p,q,\phi}(f,\Omega,X): \left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1\right\}.\end{equation*} In this article, we extensively study the Bohr radius $R_{p,q,\phi}(\Omega,X)$ , when X is an arbitrary Banach space, and $X=\mathcal{B}(\mathcal{H})$ is the algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$. Furthermore, we establish the Bohr inequality for the operator-valued Cesáro operator and Bernardi operator. [ABSTRACT FROM AUTHOR]

In this article, we study the Bohr type inequality for Cesáro operator and Bernardi integral operator acting on the space of analytic functions defined on a simply connected domain containing the unit disk D . [ABSTRACT FROM AUTHOR]

In this paper, we establish several new versions of Bohr-type inequalities for bounded analytic functions in the unit disk by allowing φ = { φ n (r) } n = 0 ∞ in place of the { r n } n = 0 ∞ in the power series representations of the functions involved with the Bohr sum and thereby introducing a single parameter, which generalize several related results of earlier authors. [ABSTRACT FROM AUTHOR]

The concept of the Bohr radius of a pair of Banach spaces is introduced. The lower estimate for the value of the Bohr radius from the Bloch space to the space of bounded functions obtained by I. Kayumov, S. Ponnusamy and N. Shakirov is slightly improved. It is shown that for any weighted Bloch space the Bohr radius is not less than . A criterion for the sharpness of the Bohr inequality in the weighted Bloch space with is obtained. Using this criterion, the examples of the weights for which the inequality is sharp is given. [ABSTRACT FROM AUTHOR]

Bohr's classical theorem and its generalizations are now active areas of research and have been the source of investigations in numerous function spaces. In this article, we study a generalized Bohr inequality for the class of bounded analytic functions defined on the simply connected domain Ω γ := { z ∈ C : | z + γ 1 − γ | < 1 1 − γ } , f o r 0 ≤ γ < 1. Part of its applications, we calculate the Bohr-type radii for some known integral operators. [ABSTRACT FROM AUTHOR]

In order to improve the classical Bohr inequality, we explain some refined versions for a quasi‐subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk D${\mathbb {D}}$. Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form f=h+g¯$f=h+\overline{g}$, where g(0)=0$g(0)=0$, the analytic part h is bounded by 1 and that |g′(z)|≤k|h′(z)|$|g^{\prime }(z)|\le k|h^{\prime }(z)|$ in D${\mathbb {D}}$ and for some k∈[0,1]$k\in [0,1]$. [ABSTRACT FROM AUTHOR]

In this paper, first we prove two refined Bohr-type inequalities associated with area for bounded analytic functions f (z) = ∑ n = 0 ∞ a n z n in the unit disk. Later, we establish the Bohr-type operator on analytic functions and sections. [ABSTRACT FROM AUTHOR]

Let H be the class of complex-valued harmonic mappings f = h + g ¯ defined in the unit disk D := { z ∈ C : | z | < 1 } , where h and g are analytic functions in D with the normalization h (0) = 0 = h ′ (0) − 1 and g (0) = 0. Let H 0 = { f = h + g ¯ ∈ H : g ′ (0) = 0 }. Let P H 0 (M) := { f = h + g ¯ ∈ H 0 : Re (z h ′ ′ (z)) > − M + | z g ′ ′ (z) | , z ∈ D and M > 0 }. be the class of harmonic univalent mappings in the unit disk D [Ghosh N, Allu V. On some subclasses of harmonic mappings. Bull Aust Math Soc. 2020;101:130–140.]. In this paper, we obtain the sharp Bohr–Rogosinski inequality, improved Bohr inequality, refined Bohr inequality and Bohr-type inequality for the class P H 0 (M). [ABSTRACT FROM AUTHOR]

By the principle of differential subordination and the $ q $-derivative operator, we introduce the $ q $-analog $ \mathcal{SP}^{q}_{snail}(\lambda; \alpha, \beta, \gamma) $ of certain class of analytic functions associated with the generalized Pascal snail. Firstly, we obtain the coefficient estimates and Fekete-Szegö functional inequalities for this class. Meanwhile, we also estimate the corresponding symmetric Toeplitz determinant. Secondly, for all the above results we provide the corresponding results for the reduced classes $ \mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma) $ and $ \mathcal{RP}^{q}_{snail}(\alpha, \beta, \gamma) $. Thirdly, we characterize the Bohr radius problems for the function class $ \mathcal{SP}^{q}_{snail}(\alpha, \beta, \gamma) $. Lastly, we establish certain results for some new subclasses of functions defined by the neutrosophic Poisson distribution series.

Let denote the class of all analytic functions mapping the unit disk into itself. We consider the powered Bohr sum with an additional term including the area of the Riemann surface which is the image of the disk . In this setting, we prove the counterpart of the Bohr theorem for two cases, considering different values of the zero coefficient . [ABSTRACT FROM AUTHOR]

In this paper, some new versions of Bohr-type inequalities for bounded analytic functions of Schwarz functions are established. Most of these inequalities are sharp. Some previous inequalities are generalized.

In Geometric function theory, occasionally, attempts have been made to solve a particular problem for the Ma-Minda classes, S ∗ (ψ) and C (ψ) of univalent starlike and convex functions, respectively. Recently, a popular radius problem generally known as Bohr’s phenomenon has been studied in various settings; however, a little is known about Rogosinski radius. In this article, for a fixed f ∈ S ∗ (ψ) or C (ψ) , the class of analytic subordinants S f (ψ) : = { g : g ≺ f } is studied for the Bohr–Rogosinski phenomenon in a general setting. Its applications to the classes S ∗ (ψ) and C (ψ) are also shown. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the classical Bohr sum for analytic functions defined on the unit disk in a general setting. In addition, we discuss a generalization of the Bohr–Rogosinski sum for a class of analytic functions subordinate to the univalent functions in the unit disk. Several well-known results are observed from the consequences of our main results. [ABSTRACT FROM AUTHOR]

Let H be the class of complex-valued harmonic mappings f = h + g ¯ defined in the unit disk D , where h and g are analytic functions in D with h (0) = 0 = h ′ (0) - 1 and g (0) = 0 and H 0 = { f = h + g ¯ ∈ H : g ′ (0) = 0 } and P H 0 (M) : = { f = h + g ¯ ∈ H 0 : Re (z h ″ (z)) > - M + | z g ″ (z) | , z ∈ D , M > 0 } . Functions in the class P H 0 (M) are known to be fully starlike univalent functions for 0 < M < 1 / log 4 . In this paper, we obtain the sharp Bohr–Rogosinski type inequality and improved Bohr inequality and the corresponding Bohr radius for the class P H 0 (M) . [ABSTRACT FROM AUTHOR]

For any K > 2 and the multiplicative cyclic group Ω K of order K , consider any function f : Ω K n → C and its Fourier expansion f (z) = ∑ α ∈ { 0 , 1 , ... , K − 1 } n a α z α , with d : = deg ⁡ (f) denoting its degree as a multivariate polynomial. We prove a Bohnenblust–Hille (BH) inequality in this setting: the ℓ 2 d / (d + 1) norm of the Fourier coefficients of f is bounded by C (d , K) ‖ f ‖ ∞ with C (d , K) independent of n. This is the interpolating case between the now well-understood BH inequalities for functions on the poly-torus (K = ∞) and the hypercube (K = 2) but those extreme cases of K have special properties whose absence for intermediate K prevent a proof by the standard BH framework. New techniques are developed exploiting the group structure of Ω K n. By known reductions, the cyclic group BH inequality also entails a noncommutative BH inequality for tensor products of the K × K complex matrix algebra (or in the language of quantum mechanics, systems of K -level qudits). These new BH inequalities generalize several applications in harmonic analysis and statistical learning theory to broader classes of functions and operators. [ABSTRACT FROM AUTHOR]

Let HC (ϕ) and HC c (ϕ) denote the classes of sense-preserving harmonic mappings f = h + g ¯ in D with the dilation g ′ (z) = α z h ′ (z) for | α | < 1 such that h is Ma–Minda type convex function and Ma–Minda type convex function with respect to the conjugate points respectively. Here, the function ϕ : D → ℂ , called Ma–Minda function, is analytic and univalent in D such that ϕ (D) has positive real part, symmetric with respect to the real axis, starlike with respect to ϕ (0) = 1 and ϕ ′ (0) > 0. The classes HC (ϕ) and HC c (ϕ) are derived from the work of Sun et al. (2016) on the class M (α , β). We study the growth theorems for the functions in these classes. For general ϕ , the sharp coefficient bounds of the functions in the classes HC (ϕ) and HC c (ϕ) are not yet known. For the function f = h + g ¯ of the form h (z) = z + ∑ n = 2 ∞ a n z n and g (z) = ∑ n = 2 ∞ b n z n in D , using the Bohr phenomenon for subordination classes, we find the radius R f < 1 such that Bohr inequality | z | + ∑ n = 2 ∞ (| a n | + | b n |) | z | n ≤ d (f (0) , ∂ f (D)) holds for | z | = r ≤ R f for functions in the classes HC (ϕ) and HC c (ϕ) . As a consequence, we obtain several interesting corollaries on Bohr inequality for the aforesaid classes. In addition, we discuss the area theorem for functions in the class HC (ϕ) and obtain an improved version of Bohr inequality for functions in HC (ϕ). [ABSTRACT FROM AUTHOR]

In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr’s inequalities for a subclass of close-to-convex harmonic mappings, whose analytic parts are Ma-Minda convex functions. [ABSTRACT FROM AUTHOR]

We extend a classical result of Bombieri [8] for operator-valued holomorphic functions defined in the open unit disk D . [ABSTRACT FROM AUTHOR]

The main aim of this paper is to prove a generalization of the classical Bohr theorem and as an application, we obtain a counterpart of Bohr theorem for the generalized Cesáro operator. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Bohr phenomenon for the class of analytic functions defined on the simply connected domain S = < z C: z 4--< -- > for O = 7 < 1. ' I I 1 -7| 1 -7 J We study improved Bohr radius, Bohr Rogosinski radius and refined Bohr radius for the class of analytic functions defined in O·, and obtain several sharp results. [ABSTRACT FROM AUTHOR]

In this article, some Bohr-type inequalities with one parameter or involving convex combination for bounded analytic functions of Schwarz functions are established. Some previous inequalities are generalized. All the results are sharp. [ABSTRACT FROM AUTHOR]

In this paper we obtain the sharp Bohr radius, Bohr-Rogosinski radius, improved Bohr-radius and refined Bohr radius for the functions in the class G0/H (γ) of Goodman-Ronning type harmonic univalent functions with negative coeffcients. [ABSTRACT FROM AUTHOR]

Due to their tunable properties, silicon nano-crystals (NC) are currently being investigated. Quantum confinement can generally be employed for size-dependent band-gap tuning at dimensions smaller than the Bohr radius (∼5 nm for silicon). At the nano-meter scale, however, increased surface-to-volume ratio makes the surface effects dominant. Specifically, in Si-SiO2 core-shell semiconductor NCs the interfacial transition layer causes peculiar electronic and optical properties, because of the co-existence of intermediate oxidation states of silicon (Sin+, n=0-4). Due to the presence of the many factors involved, a comprehensive understanding of the optical properties of these NCs has not yet been achieved. In this work, Si-SiO2 NCs with a diameter of 1.1 nm and covered by amorphous oxide shells with thicknesses between 2.5 and 4.75 Å are comprehensively studied, employing density functional theory calculations. It is shown that with increased oxide shell thickness, the low-energy part of the optical transition spectrum of the NC is red shifted and attenuated. Moreover, the absorption coefficient is increased in the high-energy part of the spectrum which corresponds to SiO2 transitions. Structural examinations indicate a larger compressive stress on the central silicon cluster with a thicker oxide shell. Examination of the local density of states reveals the migration of frontier molecular orbitals from the oxide shell into the silicon core with the increase of silica shell thickness. The optical and electrical properties are explained through the analysis of the density of states and the spatial distribution of silicon sub-oxide species. [ABSTRACT FROM AUTHOR]

Let H be the class of harmonic functions f = h + g ¯ in the unit disk D : = { z ∈ C : | z | < 1 } , where h and g are analytic in D . Let P H 0 (α) = { f = h + g ¯ ∈ H : Re (h ′ (z) - α) > | g ′ (z) | with 0 ≤ α < 1 , g ′ (0) = 0 , z ∈ D } be the class of close-to-convex mappings defined by Li and Ponnusamy (Nonlinear Anal 89:276–283, 2013). In this paper, we obtain the sharp Bohr–Rogosinski radius, improved Bohr radius and refined Bohr radius for the class P H 0 (α) . [ABSTRACT FROM AUTHOR]

The electronic spin and valley transport properties of a monolayer MoS2 are investigated using the non-equilibrium Green's function formalism combined with density functional theory. Due to the presence of strong Rashba spin orbit interaction (RSOI), the electronic valence bands of monolayer MoS2 are split into spin up and spin down Zeeman-like texture near the two inequivalent vertices K and K' of the first Brillouin zone. When the gate voltage is applied in the scattering region, an additional strong RSOI is induced which generates an effective magnetic field. As a result, electron spin precession occurs along the effective magnetic field, which is controlled by the gate voltage. This, in turn, causes the oscillation of conductance as a function of the magnitude of the gate voltage and the length of the gate region. This current modulation due to the spin precession shows the essential feature of the long sought Datta-Das field effect transistor (FET). From our results, the oscillation periods for the gate voltage and gate length are found to be approximately 2.2V and 20.03aB (aB is Bohr radius), respectively. These observations can be understood by a simple spin precessing model and indicate that the electron behaviors in monolayer MoS2 FET are both spin and valley related and can easily be controlled by the gate. [ABSTRACT FROM AUTHOR]

In this paper, we shall investigate some generalized Bohr radius in the non-commutative setting of quaternions. To be precise, two results of Enrico Bombieri are generalized by means of a new idea into the class of slice regular functions over quaternions. As an application, the Bohr radius is determined for p-symmetric slice regular functions over quaternions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a new subclass of harmonic mappings which satisfy a third-order differential inequality. Such results as close-to-convexity, coefficient bounds, growth estimates, sufficient coefficient condition and convolution properties are derived. Furthermore, we obtain several improved versions of the sharp Bohr radius for harmonic mappings. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the Bohr-type radii for several different forms of Bohr-type inequalities of analytic functions in the unit disk, we also investigate the Bohr-type radius of the alternating series associated with the Taylor series of analytic functions. We will prove that most of the results are sharp.

In this paper, we first establish a version of multidimensional analogues of the refined Bohr's inequality. Then we establish two versions of multidimensional analogues of improved Bohr's inequality with initial coefficient being zero. Finally we establish two versions of multidimensional analogues of improved Bohr's inequality with the initial coefficient being replaced by absolute value of the function, and to prove that most of the results are sharp. [ABSTRACT FROM AUTHOR]

The Bohr radius for a class G consisting of analytic functions f (z) = ∑ n = 0 ∞ a n z n in unit disc D = { z ∈ C : | z | < 1 } is the largest r ∗ such that every function f in the class G satisfies the inequality d ∑ n = 0 ∞ | a n z n | , | f (0) | = ∑ n = 1 ∞ | a n z n | ≤ d (f (0) , ∂ f (D)) for all | z | = r ≤ r ∗ , where d is the Euclidean distance. In this paper, our aim is to determine the Bohr radius for the classes of analytic functions f satisfying differential subordination relations z f ′ (z) / f (z) ≺ h (z) and f (z) + β z f ′ (z) + γ z 2 f ′ ′ (z) ≺ h (z) , where h is the Janowski function. Analogous results are obtained for the classes of α -convex functions and typically real functions, respectively. All obtained results are sharp. [ABSTRACT FROM AUTHOR]

The key purpose of this paper is to investigate the Bohr radius for several subclasses of analytic functions with negative coefficients. Our investigation with the Bohr radius correlates with the classes of generalized Janowski type functions. Under this novel strategy, we develop Bohr's phenomenon for a generalized class associated with q-functions having q 2 (0; 1). In the applications viewpoint, our consequences have shown the applicability in the class inaugurated by Bessel functions. [ABSTRACT FROM AUTHOR]