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This article comprises the study of strongly starlike functions which are defined by using the concept of subordination. The function φ defined by φ(ζ)=(1+ζ)λ, 0

The main goal of this investigation is to obtain sharp upper bounds for Fekete-Szegö functional and the third Hankel determinant for a certain subclass SL∗u,v,α of holomorphic functions defined by the Carlson-Shaffer operator in the unit disk. Finally, for some special values of parameters, several corollaries were presented.

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.

In this paper, we introduce and study the class of analytic functions in the unit disc, which are derived from Robertson’s analytic formula for starlike functions with respect to a boundary point combined with a subordination involving lemniscate of Bernoulli and crescent shaped domains. Using their symmetry property, the basic geometrical and analytical properties of the introduced classes were proved. Early coefficients and the Fekete–Szegö functional were estimated. Results for both classes were also obtained by applying the theory of differential subordinations.

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.

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-Szego for this class is evaluated. Also some geometric results related to subordinations are evaluated for this class in connection with Janowski functions.

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.

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

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.

Sufficient conditions on associated parametersp, bandcare obtained so that the generalized and “normalized” Bessel functionup(z)=up,b,c(z)satisfies the inequalities ∣(1 + (zu″p(z)/u′p(z)))2− 1∣ < 1 or ∣((zup(z))′/up(z))2− 1∣ < 1. We also determine the condition on these parameters so that . Relations between the parametersμandpare obtained such that the normalized Lommelfunction of first kindhμ,p(z)satisfies the subordination . Moreover, the properties of Alexander transform of the functionhμ,p(z) are discussed.

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&ndash, Szeg&ouml, problem.

This 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.

Several inclusions between the class of functions with positive real part and the class of starlike univalent functions associated with lemniscate of Bernoulli are obtained by making use of the well-known theory of differential subordination. Further, these inclusions give sufficient conditions for normalized analytic functions to belong to some subclasses of starlike functions. The results also provide sharp version of some previously known results.

In this paper, a new subclass of analytic functions ML_{\lambda}^{*} associated with the right half of the lemniscate of Bernoulli is introduced. The sharp upper bound for the Fekete-Szego functional |a_{3}-\mu a_{2}^{2}| for both real and complex \mu are considered. Further, the sharp upper bound to the second Hankel determinant |H_{2}(1)| for the function f in the class ML_{\lambda}^{*} using Toeplitz determinant is studied. Relevances of the main results are also briefly indicated.

The main aim of this manuscript is to investigate sharp bound on the functional $$|a_{p+1}a_{p+2}-a_{p+3}|$$ for functions $$f(z)=z^p+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+a_{p+3}z^{p+3}+\cdots $$ belonging to the class $$\mathcal {R}_p(\alpha )$$ associated with the right half-plane. Also sharp bounds on the initial coefficients, bounds on $$|a_{p+1}a_{p+2}-a_{p+3}|$$ , and $$|a_{p+1}a_{p+3}-a_{p+2}^2|$$ for functions in the class $$\mathcal {RL}_p(\alpha )$$ , related to the lemniscate of Bernoulli, are also derived. Further, these estimates are used to derive a bound on the third Hankel determinant.

Radii of convexity, starlikeness, lemniscate starlikeness and close-to-convexity are determined for the convex combination of the identity map and a normalized convex functionFgiven byf(z) =α z+(1−α)F(z).

For an analytic function $f$ on the unit disk $\mathbb{D}=\{z:|z, 20 pages

Let $$\mathcal {S}^*_C$$ be the class of normalized analytic functions f in the unit disc with $$zf'(z)/f(z)$$ lying in the region bounded by the cardioid given by the equation $$(9 x^2+9 y^2-18x+5)^2- 16 (9 x^2+9 y^2-6x+1)=0$$ . We determine the structural formula, coefficient estimates, growth results and various radii constants such as the radius of starlikeness, radius of lemniscate of Bernoulli starlikeness, radius of M-starlikeness and radius of $$\mathcal {M}(\beta )$$ -starlikeness for functions in the class $$\mathcal {S}^*_C$$ . In addition, the $$\mathcal {S}^*_C$$ -radii for functions belonging to several interesting classes are determined. All the results obtained are sharp.

We obtain the conditions onso that 1+�zp ' (z) ≺ 1+4z/3+ 2z 2 /3 implies p(z) ≺ (2+z)/(2−z), 1+(1−�)z, (1+(1−2�)z)/(1−z), (0 ≤ � < 1), exp(z) or √ 1 + z. Similar results are obtained by considering the expressions 1+�zp ' (z)/p(z), 1+�zp ' (z)/p 2 (z) and p(z)+�zp ' (z)/p(z). These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition |log(zf ' (z)/f(z))| < 1 or |(zf ' (z)/f(z)) 2 −1| < 1 or zf ' (z)/f(z) lying in the region bounded by the cardioid (9x 2 + 9y 2 − 18x + 5) 2 − 16(9x 2 + 9y 2 − 6x + 1) = 0.

The object of the present paper is to derive the radius of convexity for certain analytic functions associated with the lemniscate of Bernoulli.

Motivated by a problem posed by David A. Singer in 1999 and by the classical Bernoulli lemniscate and the Norwich spiral, we study the plane curves whose curvature is expressed in terms of the distance from a point. In terms of the geometric angular momentum, we provide new characterizations of some well-known curves, like the mentioned Bernoulli lemniscate, the inverse Norwich spiral, the anti-clothoid, the cardioid, the sinusoidal spirals and the Cassini ovals. We also find out several new families of plane curves whose intrinsic equations are expressed in terms of elementary functions and we are able to get their arc-length parametrizations and they are depicted graphically.

Let \(\fancyscript{S}_{e}^*\) denote the class of analytic functions \(f\) in the open unit disk normalized by \(f(0)=f'(0)-1=0\) and satisfying the condition \(zf'(z)/f(z)\prec e^z\) for \(|z

In this article, we present purely geometrical proofs of the well-known properties of the lemniscate of Bernoulli.

In this paper, our aim is to define a new subclass of close-to-convex functions in the open unit disk U that are related with the right half of the lemniscate of Bernoulli. For this function class, we obtain the upper bound of the third Hankel determinant. Various other related results are also considered.

The area of [Formula: see text]-calculus has attracted the serious attention of researchers. This great interest is due its application in various branches of mathematics and physics. The application of [Formula: see text]-calculus was initiated by Jackson [Jackson, On [Formula: see text]-definite integrals, Quart. J. Pure Appl. Math. 41 (1910) 193–203; On [Formula: see text]-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253–281.], who was the first to develop [Formula: see text]-integral and [Formula: see text]-derivative in a systematic way. In this paper, we make use of the concept of [Formula: see text]-calculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of [Formula: see text]-starlike and [Formula: see text]-convex functions. Further, we also obtain similar type of inequalities related to lemniscate of Bernoulli. The authors sincerely hope that this paper will revive this concept and encourage other researchers to work in this [Formula: see text]-calculus in the near-future in the area of complex function theory. Also, we present a direct and shortened proof for the estimates of [Formula: see text] found in [Mishra and Gochhayat, Fekete–Szego problem for [Formula: see text]-uniformly convex functions and for a class defined by Owa–Srivastava operator, J. Math. Anal. Appl. 347(2) (2008) 563–572] for [Formula: see text], [Formula: see text].

Several radii problems are considered for functions f (z) = z + a2z2 + ... with fixed second coefficient a2. For 0 ? ? < 1, sharp radius of starlikeness of order ? for several subclasses of functions are obtained. These include the class of parabolic starlike functions, the class of Janowski starlike functions, and the class of strongly starlike functions. Sharp radius of convexity of order ? for uniformly convex functions, and sharp radius of strong-starlikeness of order ? for starlike functions associated with the lemniscate of Bernoulli are also obtained as special cases.

Thirteen simple closed geodesics are found in the lemniscate. Among these are nine “mirrors”—geodesics of reflection symmetry—which generate the full octahedral group and determine a triangulation of the lemniscate as a disdyakis dodecahedron. New visualizations of the lemniscate are presented.

In this paper we consider some coe-cient estimates in the subclass SL ⁄ of strongly starlike functions deflned by a certain geometric condition.

Several subclasses of starlike functions are associated with regions in the right half plane of the complex plane, like half-plane, disks, sectors, parabolas and lemniscate of Bernoulli. For a normalized analytic function [Formula: see text] defined on the open unit disk [Formula: see text] belonging to certain well-known classes of functions associated with the above regions, we investigate the radius [Formula: see text] such that, for the function [Formula: see text], [Formula: see text] lies in the lune defined by [Formula: see text] for all [Formula: see text].

Conditions on $\beta$ are determined so that $1+\beta zp'(z) $ subordinated to $ \sqrt{1+z} $ implies $ p $ is subordinated to $ \sqrt{1+z}$. Analogous results are also obtained involving the expressions $1+\beta{zp'(z)}/{p(z)}$ and $1+\beta{zp'(z)}/{p^2(z)}$. These results are applied to obtain sufficient conditions for normalized analytic functions $f$ to satisfy the condition $|\left({zf'(z)}/{f(z)}\right)^2-1

For fixed integersk≥2 we study sequences of polynomialsP n (z) with the following properties: (i) degP n → ∞; (ii) the zeros of all theP n (z) lie on a certain lemniscate withk 1≤k foci, one of which is the origin; (iii) theP n (z) can be cut in such a way that the zeros of the “lower part” all lie on the unit circle and those of the “upper part” lie on a lemniscate having the foci in (ii) excluding the origin. Several special cases and examples are considered.

Let $\mathcal{S}_{RL}^{*}$ denote the class of all analytic functions f in the open unit disk with the normalizations f(0) = f′(0) - 1 = 0 such that zf′(z)/f(z) lies in the interior of the left-half of the shifted lemniscate of Bernoulli $((x-\sqrt{2})^2+y^2)^2-2((x-\sqrt{2})^2-y^2) = 0$. In this paper, we investigate the geometric properties of functions in this class and compute the $\mathcal{S}_{RL}^*$-radius for functions belonging to several interesting classes.

We consider complex curves of genus 0 and answer the above riddle. Namely, the lemniscate of Bernoulli, which has obvious fourfold symmetry, actually has the octahedral group as its symmetr...

is known as the lemniscate constant. Here B(x, y) and V(z) are the beta and gamma functions, respectively, whose definitions we recall in the next section. The lemniscate constant gets its name from the fact that the arclength of the lemniscate with polar equation r2 = cos(2#) is given by 2L, just as the arclength of the unit circle is given by 2tt. In our work, the lemniscate constant will arise in the guise of an area. We will see that the area enclosed by the curve C4 defined by x4 + y4 ? 1 is La/2, and this will allow us to give a geometric meaning to the product formula (2). For more details on the remarkable lemniscate constant, we refer the reader to [11, pp. 420-423]. Equation (1) is classical. It was discovered in 1593 by Fran?ois Vi?te. It was the first exact analytic expression ever discovered for tt and constitutes the first known use of an infinite product in mathematics. Vi?te discovered the formula by considering the areas of regular 2"-gons inscribed in a unit circle. For Vi?te's original paper, in Latin or translated into English, see [5]. For a discussion of Vi?te's product and its place in the history of mathematics and tt see [4, pp. 92-96]. Equation (2) appeared in [17] as a consequence of a general method for constructing similar infinite product formulas. We will show that the similarity between equations (1) and (2) goes beyond mere typographical appearances. We will see that (2) is related to the curve C4 in much the same way that Vi?te's product is related to the circle.