Lemniscate Of Bernoulli Research Articles

Upper Bound of the Third Hankel Determinant for a Subclass of Multivalent Functions Associated with the Bernoulli Lemniscate

Let RL; SL and CL represent the families of multivalent bounded turning, multivalent starlike, and multivalent convex functions that are subordinated with Bernoulli lemniscate in the open unit disk E = {z : |z|<1}. In this particular paper, our goal is to find the upper bounds of Hankel’s third order determinant for the above-mentioned families.

Radius of Star-Likeness for Certain Subclasses of Analytic Functions

In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition ℜfzgz>0, for some analytic function, g, with ℜz+1−2nzgz>0,∀n∈N. We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp.

On q-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

This article comprises the study of differential subordination with analogue of q -derivative. It includes the sufficient condition on γ for 1 + γ ∂ z q h z / h n z to be subordinated by 1 + A z / 1 + B z , − 1 ≤ B < A ≤ 1 , and implies that h z ≺ 1 + z , where h z is the analytic function in the open unit disk. Moreover, certain sufficient conditions for q -starlikeness of analytic functions related with lemniscate of Bernoulli are determined.

Starlikeness of certain non-univalent functions

We consider three classes of functions defined using the class {mathcal {P}} of all analytic functions p(z)=1+cz+cdots on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions f with f/gin {mathcal {P}} and g/(zp)in {mathcal {P}} for some normalized analytic function g and pin {mathcal {P}}. The second class is defined by replacing the condition f/gin {mathcal {P}} by |(f/g)-1|<1 while the other class consists of normalized analytic functions f with f/(zp)in {mathcal {P}} for some pin {mathcal {P}}. We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order alpha , parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.

Radius of starlikeness of certain analytic functions

We investigate radius problems for three classes of normalized analytic functions f characterized by its ratio with certain normalized analytic functions g. The first two classes consist of functions f satisfying $${\text {Re}} (f(z)/g(z))>0$$ with either $${\text {Re}} (g(z)/s(z))>0$$ or $${\text {Re}} (g(z)/s(z))>1/2$$ and the other class consists of functions f satisfying $$\left|(f(z)/g(z))-1\right|<1$$ with $${\text {Re}} (g(z)/s(z))>0$$ for some starlike function s of order $$\alpha $$ , $$0\le \alpha <1$$ . For functions in these classes, we compute the various radii of starlikeness, including the radius of starlikeness of order $$\alpha $$ , and radius of Ma-Minda starlikeness associated with parabola, lemniscate of Bernoulli, exponential function, cardioid, sine, lune, rational function and nephroid.

Properties of Certain Multivalent Analytic Functions Associated with the Lemniscate of Bernoulli

Using differential subordination, we consider conditions of β so that some multivalent analytic functions are subordinate to (1+z)γ (0&lt;γ≤1). Notably, these results are applied to derive sufficient conditions for f∈A to satisfy the condition zf′(z)f(z)2−1&lt;1. Several previous results are extended.

Sufficiency Criteria for q -Starlike Functions Associated with Cardioid

This article deals with the q -differential subordinations for starlike functions associated with the lemniscate of Bernoulli and cardioid domain. The primary goal of this work is to find the conditions on γ for 1 + γ z ∂ q h z / h n z ≺ 1 + z , where h z is analytic function and is subordinated by the function which is producing cardioid domain as its image domain while mapping the open unit disk. Along with this, certain sufficient conditions for q -starlikeness of analytic functions are determined.

Radius of starlikeness of certain analytic functions

AbstractThis paper studies analytic functionsfdefined on the open unit disk of the complex plane for whichf/gand (1 +z)g/zare both functions with positive real part for some analytic functiong. We determine radius constants of these functions to belong to classes of strong starlike functions, starlike functions of orderα, parabolic starlike functions, as well as to the classes of starlike functions associated with lemniscate of Bernoulli, cardioid, lune, reverse lemniscate, sine function, exponential function and a particular rational function. The results obtained are sharp.

On $ q $-analogue of meromorphic multivalent functions in lemniscate of Bernoulli domain

Utilizing the concepts from $ q $-calculus in the field of geometric function theory, we introduce a subclass of $ p $-valent meromorphic functions relating to the domain of lemniscate of Bernoulli. The well known problem of Fekete-Szegö for this class is evaluated. Also some geometric results related to subordinations are evaluated for this class in connection with Janowski functions.

SOME GEOMETRIC PROPERTIES OF MULTIVALENT ANALYTIC FUNCTIONS ASSOCIATED WITH THE LEMNISCATE OF BERNOULLI

<abstract> By using the first-order differential subordination, a new class $ \mathcal{M}_n(\alpha) $ of multivalent analytic functions associated with the lemniscate of Bernoulli is introduced. Several geometric properties of this class are given. </abstract>

On Certain Class of Bazilevič Functions Associated with the Lemniscate of Bernoulli

Making use of the principle of subordination, we introduce a certain class of multivalently Bazilevi c ˘ functions involving the Lemniscate of Bernoulli. Also, we obtain subordination properties, inclusion relationship, convolution result, coefficients estimate, and Fekete–Szegö problem for this class.

Multiplication and Division on Elliptic Curves, Torsion Points, and Roots of Modular Equations

Having expressed the ratio of the length of the lemniscate of Bernoulli to the length of its cocentered superscribing circle as the reciprocal of the arithmetic-geometric mean of 1 and $$ \sqrt{2} $$ , Gauss wrote in his diary, on May 30, 1799, that thereby “an entirely new field of analysis” emerged. Yet, up to these days, the study of elliptic functions (and curves) has been based on two traditional approaches (namely, that of Jacobi and that of Weiestrass), rather than a single unifying approach. Replacing artificial dichotomy by a methodologically justified single unifying approach not only enables rederiving classical results eloquently, but allows one to undertake new calculations, which seemed either unfeasible or too cumbersome to be explicitly performed. Here, we shall derive readily verifiable explicit formulas for carrying out highly efficient arithmetic on complex projective elliptic curves. We shall explicitly relate calculating the roots of the modular equation of level p to calculating the p-torsion points on a corresponding elliptic curve, and we shall rebring to light Galois’ exceptional, never really surpassable, and far from fully appreciated impact. Bibliography: 19 titles.

On fourth Hankel determinant for functions associated with Bernoulli's lemniscate

The aim of this paper is to find an upper bound of the fourth Hankel determinant $H_{4}(1)$ for a subclass of analytic functions associated with the right half of the Bernoulli's lemniscate of the form $\left(x^{2}+y^{2}\right) ^{2}-2\left( x^{2}-y^{2}\right) =0$. The problem is also discussed for 2-fold and 3-fold symmetric functions. The key tools in the proof of our main results are the coefficient inequalities for class $\mathcal{P}$ of functions with positive real part.***************************************************************************************************************************

The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*

Abstract In this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H 3(1) and H 2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: | a 3 2 − a 5 | $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

Curves in the Lorentz-Minkowski plane with curvature depending on their position

Abstract Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in L 2 {{\mathbb{L}}}^{2} whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic), respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper surface of second kind and for Lorentzian versions of some well-known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.

Alternative Constructs of the Lemniscate of Bernoulli

Until now there are at least three alternative methods of constructing lemniscate of Bernoulli. In this paper six alternative methods will be given to construct Lemniscate of Bernoulli in a simple way. Furthermore, at the end of the session will be given how to construct the incircle of Lemniscate of Bernoulli.

Radii of Starlikeness and Convexity of Some Entire Functions

A normalized analytic function f is lemniscate starlike if the quantity $$zf'(z)/f(z)$$ lies in the region bounded by the right half of the lemniscate of Bernoulli $$|w^2-1|=1$$ . It is Janowski starlike if the quantity $$zf'(z)/f(z)$$ lies in the disk whose diametric end points are $$(1-A)/(1-B)$$ and $$(1+A)/(1+B)$$ for $$-1\le B<A\le 1$$ . The radii of lemniscate starlikeness and Janowski starlikeness have been determined for normalizations of q-Bessel functions, Bessel functions of first kind of order $$\nu $$ and Lommel functions of first kind. Corresponding convexity radii are also determined.

Some Geometric Properties of a Family of Analytic Functions Involving a Generalized q-Operator

In analysis, the introduction of q-calculus has been a revelation. It has a deep impact on various concepts and applications of pure and applied sciences. In this article we investigate certain geometric properties relating to convolution of functions of a newly defined class of analytic functions. The important region of the lemniscate of Bernoulli is considered. Here we utilize concepts of q-calculus which enhances and generalizes the vitality of this research work. In the same context we study the Fekete–Szegö problem.

On analytic multivalent functions associated with lemniscate of Bernoulli

In this paper, we establish some sufficient conditions for analytic functions associated with lemniscate of Bernoulli. In particular, we determine conditions on $\alpha $ such that $ \begin{equation*} 1+\alpha \frac{z^{2+p\left( j-1\right) }g^{\prime }\left( z\right) }{ pg^{j}\left( z\right) },\text{ for each }j = 0,1,2,3, \end{equation*} $ are subordinated by Janowski function, then $\frac{g\left(z\right) }{z^{p}} \prec \sqrt{1+z}, \left(z\in \mathfrak{D}\right) $. By choosing particular values of functions $g, $ we obtain some sufficient conditions for multivalent starlike functions associated with lemniscate of Bernoulli.

Upper bound of the third Hankel determinant for a subclass of q-Starlike functions associated with the lemniscate of Bernoulli

Upper bound of the third Hankel determinant for a subclass of q-Starlike functions associated with the lemniscate of Bernoulli