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The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order δ , k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of these conclusions to the concept of corollaries are also provided. Additionally, an illustrated representation of these outcomes will be presented. So functional inequalities involving gamma function will be the main research tools of this exploration. The outcomes from this study generalize a number of conclusions from earlier studies. [ABSTRACT FROM AUTHOR]

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ , convexity of order δ , k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature. [ABSTRACT FROM AUTHOR]

The conditions on A, B, β and γ are obtained for an analytic function p defined on the open unit disc D and normalized by p(0) = 1 to be subordinate to (1 + Az)/(1 + Bz), −1 ≤ B < A ≤ 1 when p(z) + zp' (z)/(βp(z) + γ) is subordinate to ez . The conditions on these parameters are derived for the function p to be subordinate to √ 1 + z or ez when p(z) + zp0 (z)/(βp(z) + γ) is subordinate to (1 + Az)/(1 + Bz). The conditions on β and γ are determined for the function p to be subordinate to ez when p(z)+zp'(z)/(βp(z)+γ) is subordinate to √ 1 + z. Related result for the function p(z) + zp' (z)/(βp(z) + γ) to be in the parabolic region bounded by the Re w = |w − 1| is investigated. Sufficient conditions for the Bernardi’s integral operator to belong to the various subclasses of starlike functions are obtained as applications. [ABSTRACT FROM AUTHOR]

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 < λ < 1 maps the open unit disk in the complex plane to a domain symmetric with respect to the real axis in the right-half plane. Using this mapping, we obtain some radius results for a family of starlike functions. It is worth noting that all the presented results are sharp. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we introduce the q-analogue of functions characterized by the lemniscate of Bernoulli in the right-half plane and define the class L*q(A; B). Furthermore, we study the geometric properties of this class, which include coefficient inequalities, subordination factor sequence property, radii results and Fekete-Szegö problems. Some deductions of our results show relevant connections between this present work and the existing ones in many literature. It is worthy of note that some of our results are sharp. [ABSTRACT FROM AUTHOR]

Let A consists of analytic functions f:픻→ℂ satisfying f(0)=f′(0)−1=0. Let SNe∗ be the recently introduced Ma–Minda type functions family associated with the two‐cusped kidney‐shaped nephroid curve (u−1)2+v2−493−4v23=0 given by SNe∗:=f∈A:zf′(z)f(z)≺φNe(z)=1+z−z3/3. In this paper, we adopt a novel technique that uses the geometric properties of hypergeometric functions to determine sharp estimates on β so that each of the differential subordinations p(z)+βzp′(z)≺1+z;1+z;ez;imply p (z) ≺ φNe(z), where p (z) is analytic satisfying p(0)=1. As applications, we establish conditions that are sufficient to deduce that f∈A is a member of SNe∗. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate the third and fourth Hankel determinants for the class of functions with bounded turning associated with Bernoulli's lemniscate. The fourth Hankel determinants for 2-fold symmetric and 3-fold symmetric functions are also studied. [ABSTRACT FROM AUTHOR]

Let STlim be the family of starlike functions associated with φlim (z) = 1 + √2z + z²/2, the function which sends the open unit disc ⅅ univalently onto a region bounded by the dimpled limacon curve (9u² + 9v² - 18u + 5)² - 16 (9u² + 9v² - 6u + 1) = 0. In this paper, we study certain geometrical properties along with some inclusion results related to the function class STlim.. Further, given some analytic function P{z) having nice geometrical properties, we obtain sharp estimates on the real β such that the first-order differential subordination ... implies p(z) ... φlim (z), where p is an analytic function satisfying p{0) = 1, and the symbol ... denotes subordination. As a consequence, we establish several sufficient conditions for the family STlim. [ABSTRACT FROM AUTHOR]

The object of the paper is to examine some various interseting properties of analytic multivalent functions in q-analogue associated with the lemniscate of Bernoulli. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Carathéodory function φ Ne (z) = 1 + z - z 3 / 3 maps the open unit disk D onto the interior of the nephroid, a 2-cusped kidney-shaped curve, (u - 1) 2 + v 2 - 4 9 3 - 4 v 2 3 = 0 , and introduce new Ma–Minda-type function classes S Ne ∗ and C Ne associated with it. Apart from studying the characteristic properties of the region bounded by this nephroid, the structural formulas, extremal functions, growth and distortion results, inclusion results, coefficient bounds and Fekete–Szegö problems are discussed for the classes S Ne ∗ and C Ne . Moreover, for β ∈ R and some analytic function p(z) satisfying p (0) = 1 , we prove certain subordination implications of the first-order differential subordination 1 + β z p ′ (z) / p j (z) ≺ φ Ne (z) , j = 0 , 1 , 2 , and obtain sufficient conditions for some geometrically defined function classes available in the literature. [ABSTRACT FROM AUTHOR]

We consider normalized analytic function f on the open unit disk for which either R e f (z) / g (z) > 0 , | f (z) / g (z) - 1 | < 1 or R e (1 - z 2) f (z) / z > 0 for some analytic function g with R e (1 - z 2) g (z) / z > 0 . We have obtained the radii for these functions to belong to various subclasses of starlike functions. The subclasses considered include the classes of starlike functions of order α , lemniscate starlike functions and parabolic starlike functions. [ABSTRACT FROM AUTHOR]

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 H3(1) and H2(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: |a32 − a5| for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed. [ABSTRACT FROM AUTHOR]

Sharp arc length of the image curve of a given line segment joining the points rit and -reit p0 < r < 1, 0 ¤ t ¤ 2q under the Janowski starlike and convex functions are derived. Further, length of image curve of the circle |z| r < 1 under the Janowski convex functions is also obtained. Several other results related to the arc length for many classes associated with lemniscate of Bernoulli and exponential functions are also discussed. Relevant connections of our results with the earlier known results are also pointed out. [ABSTRACT FROM AUTHOR]

The main aim of this manuscript is to investigate sharp bound on the functional |ap+1ap+2-ap+3| for functions f(z)=zp+ap+1zp+1+ap+2zp+2+ap+3zp+3+⋯ belonging to the class Rp(α) associated with the right half-plane. Also sharp bounds on the initial coefficients, bounds on |ap+1ap+2-ap+3|, and |ap+1ap+3-ap+22| for functions in the class RLp(α), related to the lemniscate of Bernoulli, are also derived. Further, these estimates are used to derive a bound on the third Hankel determinant. [ABSTRACT FROM AUTHOR]

Let p be an analytic function defined on the open unit disc D with p(0) = 1: The conditions on α and β are derived for p(z) to be subordinate to 1 + 4z=3 + 2z²=3 =: φC(z) when (1 - α)p(z) + αp²(z) + βzp′(z)/p(z) is subordinate to ez. Similar problems were investigated for p(z) to lie in a region bounded by lemniscate of Bernoulli |w² - 1| = 1 when the functions (1-α)p(z)+αp²(z)+βzp′(z), (1-α)p(z)+αp²(z)+ βzp′(z)=p(z) or p(z)+βzp′(z)=p²(z) are subordinates to φC(z). Related results for p to be in the parabolic region bounded by the Rew = |w - 1| are investigated. [ABSTRACT FROM AUTHOR]

Let S*R denote the class of normalized analytic functions f defined in the unit disk D satisfying the subordination ... . In this paper, the structural formula, coefficient estimates, growth, covering and rotation theorems for functions in the class S*R are determined. In addition, we compute the S*R-radius for functions belonging to several interesting classes. [ABSTRACT FROM AUTHOR]

Let 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 . In this paper, we investigate the geometric properties of functions in this class and compute the -radius for functions belonging to several interesting classes. [ABSTRACT FROM AUTHOR]

Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp radius constants are given. [ABSTRACT FROM AUTHOR]

Abstract: A normalized analytic function f defined on the open unit disk in the complex plane is in the class if zf′(z)/f(z) lies in the region bounded by the right-half of the lemniscate of Bernoulli given by ∣w 2 −1∣<1. In the present investigation, the -radii for certain well-known classes of functions are obtained. Radius problems associated with the left-half plane are also investigated for these classes. [Copyright &y& Elsevier]

This paper deals with a new subclass of univalent function associated with the right half of the lemniscate of Bernoulli. We find the upper bound of the Hankel determinant H 3 (1) for this subclass by applying the Carlson–Shaffer operator to it. The present work also deals with certain properties of this newly defined subclass, such as the upper bound of the Hankel determinant of order 3, coefficient estimates, etc. [ABSTRACT FROM AUTHOR]

Abstract: It has been recently proved that rational quadratic circles in standard Bézier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limaçon of Pascal. The extension to 3D rational curves is also tackled. [Copyright &y& Elsevier]

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

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. [ABSTRACT FROM AUTHOR]