On some coefficient estimates for a class of p-valent functions

In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

On some classes of p-valent functions involving Carlson–Shaffer operator

Let be the class of functions which are analytic in the open unit disk and having the form . Denote to be the class for all functions in that are univalent in . Then, let denote the class of bi-univalent functions in . In this paper, we obtain the second Hankel determinant for certain classes of analytic bi-univalent function which are defined by subordinations in the open unit disk . In particular, we determine the initial coefficients and and obtained the upper bound for the functional of functions in the classes of analytic bi-univalent function which are defined by subordinations in .

This work solves many of the classical extremal problems posed in the class of functions ${\Sigma _{K(\rho )}}$, the class of functions in $\Sigma$ with $K(\rho )$-quasiconformal extensions into the interior of the unit disk where $K(\rho )$ is a piecewise continuous function of bounded variation on $[r,1],0 \leq r < 1$. The approach taken is a variational technique and results are obtained through a limiting procedure. In particular, sharp estimates are given for the Golusin distortion functional, the Grunsky quadratic form, the first coefficient, and the Schwarzian derivative. Some extremal problems in ${S_{K(\rho )}}$, the subclass of functions in

Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of restrictions, which are in fact strong constraints, have been placed. We redefined our new class of functions using the three-parameter Mittag–Leffler function (Srivastava–Tomovski generalization of the Mittag–Leffler function), in order to increase the study’s adaptability. Coefficient estimates and their Fekete-Szegő inequalities are our main results. We have included a couple of examples to show the closure and inclusion properties of our defined class. Further, interesting bounds of logarithmic coefficients and their corresponding Fekete–Szegő functionals have also been obtained.

Results on a class of analytic functions with finitely many fixed coefficients related to a generalised multiplier transformation

&lt;abstract&gt;&lt;p&gt;In this article, we introduce a new class of analytic functions in the open unit disc that are closely related to functions that are starlike with respect to a boundary point. For this new class of functions, we obtain representation theorem, interesting coefficient estimates and also certain differential subordination implications involving this new class.&lt;/p&gt;&lt;/abstract&gt;

The aim of this paper is to obtain a sharp upper bound to the second Hankel determinant \(|a_{2}a_{4}-a_{3}^{2}|\) for the function \(f\) when it belongs to the class of parabolic starlike functions of order \(\alpha (0\le \alpha <1).\) Further, an upper bound for the inverse function of \(f\) to the non-linear functional \(|t_{2}t_{4}-t_{3}^{2}|\) was determined when it belongs to the same class of functions, using Toeplitz determinants.

In this paper we investigate the third Hankel determinant, H3(1), for normalized univalent functions f(z) = z + a2 z2 + ... belonging to the class of α-starlike functions denoted by Mα. This class includes two important subclasses of the family of univalent functions - starlike and convex functions denoted by S∗ and C. Our results therefore includes the special cases of the third Hankel determinants for the two classes of functions.

The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limacon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which $ \frac{\varsigma f^{\prime}(\varsigma)}{f(\varsigma)} $ and $ \frac{(\varsigma f^{\prime}(\varsigma))^{\prime}}{f^{\prime}(\varsigma)} $ map open unit disc onto region bounded by limacon are studied. Coefficients bounds, Fekete Szeg $ \ddot{ \rm{o}} $ inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.

Coefficient estimates and integral mean estimates for certain classes of analytic functions

The contribution of fractional calculus in the development of different areas of research is well known. This article presents investigations involving fractional calculus in the study of analytic functions. Riemann-Liouville fractional integral is known for its extensive applications in geometric function theory. New contributions were previously obtained by applying the Riemann-Liouville fractional integral to the convolution product of multiplier transformation and Ruscheweyh derivative. For the study presented in this article, the resulting operator is used following the line of research that concerns the study of certain new subclasses of analytic functions using fractional operators. Riemann-Liouville fractional integral of the convolution product of multiplier transformation and Ruscheweyh derivative is applied here for introducing a new class of analytic functions. Investigations regarding this newly introduced class concern the usual aspects considered by researchers in geometric function theory targeting the conditions that a function must meet to be part of this class and the properties that characterize the functions that fulfil these conditions. Theorems and corollaries regarding neighborhoods and their inclusion relation involving the newly defined class are stated, closure and distortion theorems are proved, and coefficient estimates are obtained involving the functions belonging to this class. Geometrical properties such as radii of convexity, starlikeness, and close-to-convexity are also obtained for this new class of functions.

In this paper, we introduce a subclass of analytic functions in the open unit disc. This class generalizes the class of Bazilevic functions of order α. We find arc length, coefficient bounds, coefficient difference, growth result, qth Hankel determinant of this class of functions.

Let T be the class of functions $$ f(z)=z+\sum_{n=2}^{\infty }{c}_n{z}^n $$ regular and typically real in the disk |z| < 1. Sharp estimates for the derivative f ′(r)(0 < r < 1) in terms of the value c3 and sharp estimates for the coefficient c3 in terms of f′(r) are obtained.

Let T be the class of functions $$ f(z)=z+\sum \limits_{n=2}^{\infty }{c}_n{z}^n $$ regular and typically real in the disk |z| < 1. Sharp estimates for the coefficients c4 and c5 in terms of f′(r) are obtained.