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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]

Thanks to the advent of affective computing, designing an automatic human emotion recognition system for clinical and non-clinical applications has attracted the attention of many researchers. Currently, multi-channel electroencephalogram (EEG)-based emotion recognition is a fundamental but challenging issue. This experiment envisioned developing a new scheme for automated EEG affect recognition. An innovative nonlinear feature engineering approach was presented based on Lemniscate of Bernoulli's Map (LBM), which belongs to the family of chaotic maps, in line with the EEG's nonlinear nature. As far as the authors know, LBM has not been utilized for biological signal analysis. Next, the map was characterized using several graphical indices. The feature vector was imposed on the feature selection algorithm while evaluating the role of the feature vector dimension on emotion recognition rates. Finally, the efficiency of the features on emotion recognition was appraised using two conventional classifiers and validated using the Database for Emotion Analysis using Physiological signals (DEAP) and SJTU Emotion EEG Dataset-IV (SEED-IV) benchmark databases. The experimental results showed a maximum accuracy of 92.16% for DEAP and 90.7% for SEED-IV. Achieving higher recognition rates compared to the state-of-art EEG emotion recognition systems suggest the proposed method based on LBM could have potential both in characterizing bio-signal dynamics and detecting affect-deficit disorders. [ABSTRACT FROM AUTHOR]

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.

The sharp radius of convexity of the classes consisting of normalized analytic functions f defined in the open unit disk is determined such that the condition | f ′ / g ′ - 1 | < 1 is satisfied in the unit disk for a univalent function g with the quantities z g ′ (z) / g (z) or 1 + z g ′ ′ (z) / g ′ (z) lying in the certain regions of the right-half plane. This, in turn, yields the sharp radius of convexity of the functions f satisfying z f ′ / f ≺ e z and z f ′ / f ≺ z + 1 + z 2 in the unit disk. [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 univalent function ϕ SG (z) : = 2 / (1 + e − z) maps the open unit disk | z | < 1 onto a sigmoid domain | log (w / (2 − w)) | < 1. The sharp radii constants have been computed for the classes involving ratios f / g and g / (z p) of analytic functions f , g and p defined in | z | < 1 such that p is subordinate to ϕ SG , the other two ratios are subordinate to 1 + z , e z or z + 1 + z 2 and the quantity z f ′ / f lies in a certain region of the right half plane. [ABSTRACT FROM AUTHOR]

In this article, our aim is to study analytic functions related with Salagean operator and associated with the right half of the lemniscate of Bernoulli. We find the estimates of the third Hankel determinant for new family of analytic functions. It is important to mention that our results generalize a number of existence results in the literature.

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]

We present the sharp bounds for the third Hankel determinant | H 3 , 1 (f − 1) | and Zalcman functional | t 3 2 − t 5 | of the inverse function of the familiar subfamily of starlike functions associated with the right half of lemniscate of Bernoulli. [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]

This paper contains an estimation of sharp upper bound (UB) for the Hankel determinant of specific orders concerning the n th -root transform [ f (z n) ] 1 n for a subfamily of holomorphic functions associated with the right half of Bernoulli's lemniscate. The practical tools applied in the derivation of our main results are the coefficient inequalities of the Carathéodory class . [ABSTRACT FROM AUTHOR]

In this paper, we establish the upper bounds of third and fourth Hankel determinants for a subclass of analytic functions in the open unit disc E = {z ... C : lzl < 1}, subordinated to Lemniscate of Bernoulli. Also we extend this investigation for two-fold and three-fold symmetric functions. Some earlier known results will follow as particular cases. [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]

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.

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.

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]

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.

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]

In this paper, we establish some sufficient conditions for analyticfunctions associated with lemniscate of Bernoulli. In particular, wedetermine 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.

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.

Sharp estimates on β in Briot-Bouquet differential subordination p(z) + βzp′(z)/p(z) ≺ h(z) are obtained so that its solution p is subordinate to some specific Carathéodory functions. As an application, the estimates on β are obtained so that the integral operator β-1 ∫0z ƒ¹/β(t)t-1dt maps the class of starlike functions ƒ satisfying zƒ′(z)ƒ(z) ≺ 1+z to various subclasses of class of starlike functions. Further, a sufficient condition is established for parabolic starlikeness. [ABSTRACT FROM AUTHOR]

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.

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]

A lemniscate is a curve defined by two foci, F1 and F2. If the distance between the focal points of F1 - F2 is 2a (a: constant), then any point P on the lemniscate curve satisfy the equation PF1 ⋅ PF2 = a². Jacob Bernoulli first described the lemniscate in 1694. The Fagnano discovered the double angle formula of the lemniscate (1718). The Euler extended the Fagnano's formula to a more general addition theorem (1751). The lemniscate function was subsequently proposed by Gauss around the year 1800. These insights were summarized by Jacobi as the theory of elliptic functions. A leaf function is an extended lemniscate function. Some formulas of leaf functions have been presented in previous papers; these included the addition theorem of this function and its application to nonlinear equations. In this paper, the geometrical properties of leaf functions at n = 2 and the geometric relation between the angle θ and lemniscate arc length l are presented using the lemniscate curve. The relationship between the leaf functions sleaf2 (l) and cleaf2 (l) is derived using the geometrical properties of the lemniscate, similarity of triangles, and the Pythagorean theorem. In the literature, the relation equation for sleaf2 (l) and cleaf2 (l) (or the lemniscate functions, sl (l) and cl (l)) has been derived analytically; however, it is not derived geometrically. [ABSTRACT FROM AUTHOR]

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 H3(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.

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 study, a normalized form of regular Coulomb wave function is considered. By using the differential subordinations method due to Miller and Mocanu, we determine some conditions on the parameters such that the normalized regular Coulomb wave function is lemniscate starlike and exponential starlike in the open unit disk, respectively. In additon, by using the relationship between the regular Coulomb wave function and the Bessel function of the first kind we give some conditions for which the classical Bessel function of the first kind is lemniscate and exponential starlike in the unit disk 𝔻. [ABSTRACT FROM AUTHOR]

In this paper we introduce an operator associated with a certain variation of the Bessel function J ν (z) in the unit disk. By using this operator and the method of differential subordination we obtain some properties such as convolution and radius of starlikeness of the function class Ω p (k , c , λ ; h) . [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]

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]

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 zf′(z)f(z)2−1<1. Several previous results are extended.

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

Sufficient conditions on associated parameters p, b and c are obtained so that the generalized and "normalized" Bessel function up(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 − (4 (p + (b + 1) / 2) / c) u p ' (x) ≺ 1 + z . Relations between the parameters μ and p are obtained such that the normalized Lommel function of first kind hμ,p(z) satisfies the subordination 1 + (z h μ , p ' ' (z) / h μ , q ' (z)) ≺ 1 + z . Moreover, the properties of Alexander transform of the function hμ,p(z) are 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]

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.

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 suficient conditions for normalized analytic functions to belong to some subclasses of starlike functions. The results also provide sharp version of some previously known results. [ABSTRACT FROM AUTHOR]

Some sufficient conditions are determined for certain first order differential subordinations to imply the corresponding analytic solution is subordinate to a rational, exponential, or sine function. By applying these results, we also obtain sufficient conditions for normalized analytic functions to be in certain well known subclasses of starlike functions. [ABSTRACT FROM AUTHOR]