Booth Lemniscate Research Articles

Hankel Determinants and Coefficient Estimates for Starlike Functions Related to Symmetric Booth Lemniscate

In this paper, we find Hankel determinants and coefficient bounds for a subclass of starlike functions related to Booth lemniscate. In particular, we obtain the first four sharp coefficient bounds, Hankel determinants of order two and three, and Zalcman conjecture for this class of functions.

The Booth Lemniscate Starlikeness Radius for Janowski Starlike Functions

The function $$G_\alpha (z)=1+ z/(1-\alpha z^2)$$ , $$0\le \alpha <1$$ , maps the open unit disk $$\mathbb {D}$$ onto the interior of a domain known as the Booth lemniscate. Associated with this function $$G_\alpha $$ is the recently introduced class $$\mathcal {BS}(\alpha )$$ consisting of normalized analytic functions f on $$\mathbb {D}$$ satisfying the subordination $$zf'(z)/f(z) \prec G_\alpha (z)$$ . Of interest is its connection with known classes $$\mathcal {M}$$ of functions in the sense $$g(z)=(1/r)f(rz)$$ belongs to $$\mathcal {BS}(\alpha )$$ for some r in (0, 1) and all $$f \in \mathcal {M}$$ . We find the largest radius r for different classes $$\mathcal {M}$$ , particularly when $$\mathcal {M}$$ is the class of starlike functions of order $$\beta $$ , or the Janowski class of starlike functions. As a primary tool for this purpose, we find the radius of the largest disk contained in $$G_\alpha (\mathbb {D})$$ and centered at a certain point $$a \in \mathbb {R}$$ .

New solutions for an Eshelby inclusion of arbitrary shape in a plane or two jointed half‐planes

AbstractNew analytical solutions are derived for Eshelby's problem of an inclusion of arbitrary shape undergoing uniform in‐plane eigenstrains in a plane or in one of two bonded half‐planes. The simply‐connected domain occupied by the inclusion is described by a conformal mapping function that maps the interior of the inclusion onto the interior of the unit circle in the image plane. Elementary expressions of the elastic fields in the exterior of the inclusion and the mean stress inside the thermal inclusion are obtained. Typical examples of epitrochoidal, Booth's lemniscate, hypotrochoidal and generalized Booth's lemniscate thermal inclusions are presented to demonstrate the new solutions.

A non-circular inhomogeneity near a non-elliptical inhomogeneity designed to admit an internal uniform hydrostatic stress field

We prove that it is possible to maintain an internal uniform hydrostatic stress field within a non-elliptical inhomogeneity in the neighborhood of a non-circular inhomogeneity when the surrounding matrix is subjected to uniform remote in-plane stresses. The non-circular inhomogeneity and the matrix have a common shear modulus but distinct Poisson's ratios. The given non-circular shapes of the inhomogeneity include a Booth's lemniscate, a generalized Booth's lemniscate and a cardioid. Our analysis indicates that the uniform hydrostatic field inside the non-elliptical inhomogeneity is unaffected by the existence of the nearby non-circular inhomogeneity whereas the non-elliptical shape of the inhomogeneity is caused solely by the presence of the non-circular inhomogeneity. Numerical examples are presented to demonstrate the general solutions obtained in each of the three non-circular shapes of the inhomogeneity.

SY3-2. Adipokine/Lipokine and inflammatory process in preeclampsia

We prove that it is possible to maintain an internal uniform hydrostatic stress field within a non-elliptical inhomogeneity in the neighborhood of a non-circular inhomogeneity when the surrounding matrix is subjected to uniform remote in-plane stresses. The non-circular inhomogeneity and the matrix have a common shear modulus but distinct Poisson's ratios. The given non-circular shapes of the inhomogeneity include a Booth's lemniscate, a generalized Booth's lemniscate and a cardioid. Our analysis indicates that the uniform hydrostatic field inside the non-elliptical inhomogeneity is unaffected by the existence of the nearby non-circular inhomogeneity whereas the non-elliptical shape of the inhomogeneity is caused solely by the presence of the non-circular inhomogeneity. Numerical examples are presented to demonstrate the general solutions obtained in each of the three non-circular shapes of the inhomogeneity.

Strength and stability of shells based on Booth lemniscate loaded with external pressure

Modern manufacturing processes and computational methods give the possibility to design shell structures with complex shapes. Such opportunity enables to use the structural advantages coming from their untypical geometries. In this paper the results of strength and stability analyses of a shell of revolution in the shape of Booth ovaloid are presented. The shell is proposed as an alternative for classical closed pressure hulls loaded with external pressure. The stress distribution along the meridian of the shell is provided as well as the buckling shapes, buckling loads and equilibrium paths. The influence of geometrical parameters of the lemniscate on the above mentioned results is discussed. The analysed shell is compared with the classical cylindrical pressure hulls as well as spherical shells, the advantages of the former are pointed out. Since the shape of the Booth lemniscate changes considerably depending on geometrical parameters it can be easily adopted to numerous applications.

Lemniscular [16]Cycloparaphenylene: A Radially Conjugated Figure-Eight Aromatic Molecule.

A cycloparaphenylene-based molecular lemniscate (CPPL) was obtained in a short synthesis involving masked p-phenylene equivalents. The strained figure-eight geometry of CPPL is sustained by the incorporated 9,9'-bicarbazole subunit, which also acts as a stereogenic element. The shape of the distorted [16]cycloparaphenylene nanohoop embedded in CPPL is accurately approximated with a Booth lemniscate. The structure of CPPL, investigated using NMR and Raman spectroscopic methods, revealed strain-dependent features, consistent with the variable curvature of the ring. The electronic and optical properties of CPPL combine features more characteristic of smaller cycloparaphenylenes, such as a reduced optical bandgap and red-shifted fluorescence. CPPL was resolved into enantiomers, which are configurationally stable and provide strong chiroptical responses, including circularly polarized luminescence.

Shaping of dished heads of the cylindrical pressure vessel for diminishing of the edge effect

The paper is devoted to diminishing of the edge effect in three nonstandard dished heads of a cylindrical pressure vessel subjected to internal uniform pressure. The problem of the edge effect diminishing in the joint of the dished head with the cylindrical shell is analytically and numerically studied. The meridians of the analysed dished heads as the shells of revolution are plane curves in the Cassini oval, Booth lemniscate and clothoid forms. Geometrical relationships of the middle surfaces of the dished heads are formulated. The stress state of these dished heads are analytically and numerically studied using finite element method in Ansys system. The results of the studies are compared and presented in Tables and Figures.

Some properties of analytic functions related with Booth lemniscate

Abstract The object of the present paper is to study of two certain subclass of analytic functions related with Booth lemniscate which we denote by ℬ𝒮 (α) and ℬ𝒦 (α). Some properties of these subclasses areconsidered.

Further Results for Starlike Functions Related with Booth Lemniscate

In this paper we investigate an interesting subclass mathcal {BS}(alpha ) (0le alpha <1) of starlike functions in the unit disc Delta. The class mathcal {BS}(alpha ) was introduced by Kargar et al. (Anal Math Phys, 2017. https://doi.org/10.1007/s13324-017-0187-3) which is strongly related to the Booth lemniscate. Some geometric properties of this class of analytic functions, including radius of starlikeness of order gamma (0le gamma <1), the image of f({z:|z|<r}) when fin mathcal {BS}(alpha ), an special example and estimate of bounds for mathrm{Re}{f(z)/z}, are studied.

Differential subordination and radius estimates for starlike functions associated with the Booth lemniscate

We obtain several inclusions between the class of functions with positive real part and the class of starlike univalent functions associated with the Booth lemniscate. These results are proved by applying the well-known theory of differential subordination developed by Miller and Mocanu and these inclusions give sufficient conditions for normalized analytic functions to belong to some subclasses of Ma-Minda starlike functions. In addition, by proving an associated technical lemma, we compute various radii constants such as the radius of starlikeness, radius of convexity, radius of starlikeness associated with the lemniscate of Bernoulli, and other radius estimates for functions in the class of functions associated with the Booth lemniscate. The results obtained are sharp.

On Booth lemniscate and starlike functions

Let $$\Delta $$ be the open unit disk in the complex plane and $$\mathcal {A}$$ be the class of normalized analytic functions in $$\Delta $$ . In this paper, we introduce and study the class $$\begin{aligned} \mathcal {BS}(\alpha ):=\left\{ f\in \mathcal {A}: \left( \frac{zf'(z)}{f(z)}-1\right) \prec \frac{z}{1-\alpha z^2}, \, z\in \Delta \right\} , \end{aligned}$$ where $$0\le \alpha \le 1$$ and $$\prec $$ is the subordination relation. Some properties of this class like differential subordination, coefficient estimates and Fekete–Szego inequality associated with the k-th root transform are considered.

On Booth Lemniscate and Hadamard Product of Analytic Functions

Abstract In [RUSCHEWEYH, S.-SHEIL-SMALL, T.: Hadamard product of schlicht functions and the Poyla-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119-135] the authors proved the P`olya-Schoenberg conjecture that the class of convex univalent functions is preserved under convolution, namely K ∗ K = K. They proved also that the class of starlike functions and the class of close-to-convex functions are closed under convolution with the class K. In this paper we consider similar convolution problems for some classes of functions. Especially we give a new applications of a result [SOKÓŁ, J.: Convolution and subordination in the convex hull of convex mappings, Appl. Math. Lett. 19 (2006), 303-306] on the subordinating relations in the convex hull of convex mappings under convolution. The paper deals with several ideas and techniques used in geometric function theory. Besides being an application to those results it provides interesting corollaries concerning special functions, regions and curves.

On the parameterization of patterns in the imaging atmospheric Cherenkov technique

A new method for analytical description of images (patterns) of the air showers detected by imaging atmospheric Cherenkov telescopes is proposed. It is shown that the generally accepted procedure constructs the image as a fourth-order plane curve (Booth's lemniscate). The method proposed for.m.s. a uniformly illuminated elliptical image preserving the same first- and second-order moments of the registered light disribution. Such a representation describes the angular size of image correctly and allows us to conclude that pixel size less than 0.2° is not particularly advantageous. Unified formulae for the calculation of Hillas' as well as the new parameters are presented. A comparative analysis of the approaches for separation of gamma- and proton-induced air showers is made. A new parameter is proposed which is more effective than AZWIDTH.

Stresses set up by rotation in thin elastic blades

The complex variable method has been used to find stresses set up in Booth's Lemniscate shaped thin isotropic elastic blade rotating steadily about a normal axis passing through its center. The complex potentials and the hoop stress are found in the closed form. The numerical values for the hoop stress are presented in the form of tables and graphs.

Torsion of beams whose sections are bounded by certain quartic curves

C auchy integral methods are applied to the torsion of a solid isotropic cylinder whose section is bounded by a quartic curve which is the inverse of an ellipse with respect to any point on its major or minor axis. The complex torsion function, the torsional rigidity and the shearing stresses are explicitly determined in the general case, and particular values are found to agree with those already known for Booth's lemniscate and the elliptic limacon. The distribution of shearing stress on the boundary is computed in two particular examples.