Search Results (118)

The concept of coefficient estimates on univalent functions is one of the interesting aspects explored recently by many researchers. Motivated by this direction, in this present work, we obtain the upper bounds of initial inverse coefficients and logarithmic coefficients and the upper bounds of differences between these successive coefficients related to concave univalent functions. Further, we also calculate the upper bounds of third-order Hankel, Toeplitz, and Vandermonde determinants in terms of specified coefficients connected to concave univalent functions. Full article

Axially symmetric Taylor–Couette flows of incompressible second grade fluids induced by time-dependent couples inside an infinite circular cylinder are studied under the action of an external magnetic field. The influence of the medium porosity is taken into account in the mathematical modeling. Analytical expressions for the dimensionless non-trivial shear stress and the corresponding fluid velocity were determined using the finite Hankel and Laplace transforms. The solutions obtained are new in the specialized literature and can be customized for various problems of interest in engineering practice. For illustration, the cases of oscillating and constant couples have been considered, and the steady state components of the shear stresses were presented in equivalent forms. Numerical schemes based on finite differences have been formulated for determining the numerical solutions of the proposed problem. It was shown that the numerical results based on analytical solutions and those obtained with the numerical methods have close values with very good accuracy. It was also proved that the fluid flows more slowly and the steady state is reached earlier in the presence of a magnetic field or porous medium. Full article

Let $\mathfrak{B}$ denote the class of bounded turning functions $\mathcal{F}$ analytic in the open unit disk, where the image of ${\mathcal{F}}^{\prime }\left(z\right)$ is contained in the domain $\Omega \left(z\right)=\cosh z+{\displaystyle \frac{2z}{2-z^2}}$. This article determines sharp coefficient bounds, a Fekete–Szegö-type inequality, and second- and third-order Hankel determinants for functions in the class $\mathfrak{B}$. Additionally, we obtain sharp Krushkal and Zalcman functional-type inequalities related to the logarithmic coefficient for functions belonging to $\mathfrak{B}$. Full article

In the present paper, using the q-difference operator, we introduce two classes of q-starlike functions and q-convex functions subordinate to secant hyperbolic functions. As functions in these classes have unique characteristic of missing coefficients on the second term in their analytic expansions, we define a new functional to unify the Hankel determinants with entries of the original coefficients, inverse coefficients, logarithmic coefficients, and inverse logarithmic coefficients for these functions. We obtain the sharp bounds on the new functional for functions in the two classes, and as a consequence, the best results on Hankel determinant for the starlike and convex functions subordinate to secant hyperbolic functions are given. The outcomes include some existing findings as corollaries and may help to deepen the understanding the properties of q-analogue analytic functions. Full article

In this study, the generalized $(p,q)$-derivative operator is used to define a novel class of bi-univalent functions. For this class, we define constraints on the coefficients up to $|{\ell }_5|$. The functions are analyzed using a suitable operational method, which enables us to derive new bounds for the Fekete–Szegö functional, as well as explicit estimates for important coefficients like $|{\ell }_2|$ and $|{\ell }_3|$. In addition, we establish the upper bounds of the second and third Hankel determinants, providing insights into the geometrical and analytical properties of this class of functions. Full article

Scholars from several disciplines have recently expressed interest in the field of fractional q-calculus based on fractional integrals and derivative operators. This article mathematically applies the fractional q-differential and q-integral operators in geometric function theory. The linear q-derivative operator ${\mathbf{S}}_{\mu ,\delta ,q}^{n,m}$ and subordination are used in this study to define and construct new classes of $\alpha$-convex functions associated with the cardioid domain. Additionally, this paper explores acute inequality problems for newly defined classes ${\mathcal{R}}_q^{\alpha }(a,c,m,L,P),$ of $\alpha$-convex functions in the open unit disc ${\mathcal{U}}_s$, such as initial coefficient bounds, coefficient inequalities, Fekete–Szegö problems, the second Hankel determinants, and logarithmic coefficients. The results presented in this paper are simple to comprehend and demonstrate how current research relates to earlier research. We found all of the estimates, and they are sharp. Full article

In this article, a new subclass of starlike functions is defined by using the technique of subordination and introducing a novel generalized domain. This domain is obtained by taking the composition of trigonometric $\sin e$ function and the well known curve called lemniscate of Bernoulli which is the image of open unit disc under a function $g\left(\xi \right)=\sqrt{1+\xi }$. This domain is characterized by its pleasing geometry which exhibits symmetric about the real axis. For this newly defined subclass, we investigate the sharp upper bounds for its first four coefficients, as well as the second and third order Hankel determinants. Full article

This study of Hankel and Hermitian Toeplitz determinants is one of the major areas of interest in Geometric function theory and has wide applications in the areas of signal processing and Applied Mathematics. In our present investigations, we define a new subclass of normalized analytic functions $\mathfrak{H}\left(\lambda \right)$$(\lambda \ge 0)$, defined using a subordination relation with the sine function $\mathcal{K}\left(z\right)=1+sinz$. For the class $\mathfrak{H}\left(\lambda \right)$, coefficient estimates, upper and lower bounds for the Hermitian Toeplitz determinants of second and third order are found. In addition, estimates are provided for the second and third-order Hankel determinants for the class $\mathfrak{H}\left(\lambda \right)$. Full article

The research on coefficient inequalities in various classes of univalent holomorphic functions focuses on interpreting their coefficients through the coefficients associated with Carathéodory functions. Therefore, researchers can investigate the behavior of coefficient functionals by applying the known inequalities for Carathéodory functions. This study will explore various coefficient inequalities employing the techniques developed for the previously discussed family of functions. These coefficient inequalities include the Krushkal, Zalcman, and Fekete-Szegö inequalities, along with the second and third Hankel determinants. The class of symmetric starlike functions linked with a petal-shaped domain is the primary focus of our study. Full article

Estimates bounds for Carathéodory functions in the complex domain are applied to demonstrate sharp limits for the inverse of analytic functions. Determining these values is considered a more difficult task compared to finding the values of analytic functions themselves. The challenge lies in finding the sharp estimate for the functionals. While some recent studies have made progress in calculating the sharp boundary values of Hankel determinants associated with inverse functions, the Toeplitz determinant is yet to be addressed. Our research aims to estimate the determinants of the Toeplitz matrix, which is also linked to inverse functions. We also focus on computing these determinants for familiar analytical functions (pre-starlike, starlike, convex, symmetric-starlike) while investigating coefficient values. The study also provides an improvement to the estimation of the determinants of the pre-starlike class presented by Li and Gou. Full article

The normalized analytic function ${\Phi }_N\left(z\right)=1+z-{\displaystyle \frac{z^3}{3}}$, which connects the open unit disk onto a bounded domain within the right half of a nephroid-shaped region, is associated with the bounded turning of functions denoted by $R_n$. It calculates the sharp coefficient inequalities, which include the upper bound of the third Hankel determinant and Logarithmic coefficients related to the functions of the ${\Phi }_N\left(z\right)$ class. This research mainly focuses on identifying solutions to specific coefficient-related problems for analytic functions within the domain of nephroid functions. Full article

This research investigates the second Hankel determinant for a specific class of functions associated with the Daehee polynomial. To achieve this, we introduce new subclasses of starlike functions in the context of Daehee polynomials. In complex analysis, establishing precise bounds for coefficient estimates in bi-univalent functions is essential, as these coefficients define the fundamental properties of conformal mappings. In this study, we derive sharp bounds for coefficient estimates within new subclasses of starlike functions related to Daehee polynomials, with most of the obtained limits demonstrating high accuracy. This work aims to inspire further exploration of rigorous bounds for analytic functions associated with innovative mapping domains. Full article

The study of the Hankel determinant generated by the Maclaurin series of holomorphic functions belonging to particular classes of normalized univalent functions is one of the most significant problems in geometric function theory. Our goal in this study is first to define a family of alpha-convex functions associated with modified sigmoid functions and then to investigate sharp bounds of initial coefficients, Fekete-Szegö inequality, and second-order Hankel determinants. Moreover, we also examine the logarithmic and inverse coefficients of functions within a defined family regarding recent issues. All of the estimations that were found are sharp. Full article

A novel family of bi-univalent holomorphic functions is introduced by the use of the Lindelöf principle. The upper bound of the second Hankel determinant, $H_{2,2}\left(\chi \right)$, is evaluated. Furthermore, specific results are obtained as special cases of the main conclusion. These cases coincide with certain recently obtained results and improve or enhance specific ones. Full article

The increasing penetration of power electronics, such as grid-connected inverters and active loads may cause power quality issues, which reduce the sensitivity of monitoring and control systems due to measurement noises. This article presents an optimal singular value decomposition (SVD) filtering method for grid-connected inverters to improve sampling accuracy against measurement noises. First, the principle of this proposed method is based on the Hankel matrix theory, and then the implementation process is explained, during which the relationship between the Hankel matrix dimension and noise reduction is discussed. Furthermore, the optimal singular value is analyzed and proposed to determine the reconstruction order. Then, the comparative analysis of the proposed optimal SVD filtering method and difference spectrum method is given to explain the optimal reconstruction order. Finally, simulation verifications are implemented to validate the effectiveness of the proposed filtering method, considering the Hankel matrix dimension, reconstruction order, and different signal–noise ratio (SNR). The verification results show that the proposed optimal SVD filtering method can accurately identify the sampling current of grid-connected inverters, even if severe harmonic noises and oscillation happen. The proposed method can reduce the effects of harmonic disturbance on measurement accuracy and control performance of grid-connected inverters, which can improve the robustness of grid-connected inverters. Full article