Hankel determinants of logarithmic coefficients for the class of bounded turning functions associated with Bell numbers

Let $\begin{array}{} \mathcal{S}^*_B \end{array}$ be the class of normalized starlike functions associated with a function related to the Bell numbers. By establishing bounds on some coefficient functionals for the family of functions with positive real part, we derive for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ several sharp coefficient bounds on the first six coefficients and also further sharp bounds on the corresponding Hankel determinants. Bounds on the first three consecutive higher-order Schwarzian derivatives for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ are investigated.

Addition formulas for polynomials built on classical combinatorial sequences

The sharp lower and upper estimates on the second- and third-order Hermitian–Toeplitz determinants for the classes of starlike functions associated with the modified sigmoid function and a related function, whose Taylor coefficients are the Bell numbers, are investigated. Further, the third and fourth Hankel determinants for these classes are also estimated.

The logarithmic coefficients are very essential in the problems of univalent functions theory. The importance of the logarithmic coefficients is due to the fact that the bounds on logarithmic coefficients of f can transfer to the Taylor coefficients of univalent functions themselves or to their powers, via the Lebedev–Milin inequalities; therefore, it is interesting to investigate the Hankel determinant whose entries are logarithmic coefficients. The main purpose of this paper is to obtain the sharp bounds for the second Hankel determinant of logarithmic coefficients of strongly starlike functions and strongly convex functions.

In the recent years, the study of the Hankel determinant problems have been widely investigated by many researchers. We were essentially motivated by the recent research going on with the Hankel determinant and other coefficient bounds problems. In this research article, we first considered the subclass of analytic starlike functions connected with the domain of the tangent function. We then derived the initial four sharp coefficient bounds, the sharp Fekete-Szegö inequality, and the sharp second and third order Hankel determinant for the defined class. Also, we derived sharp estimates like sharp coefficient bounds, Fekete-Szegö estimate, and sharp second order Hankel determinant for the functions having logarithmic coefficient and for the inverse coefficient, respectively, for the defined functions class.

The present extensive study is focused to find estimates for the upper bounds of the Toeplitz determinants. The logarithmic coefficients of univalent functions play an important role in different estimates in the theory of univalent functions, and in the this paper we derive the estimates of Toeplitz determinants and Toeplitz determinants of the logarithmic coefficients for the subclasses Ls𝑆𝑝𝑞 L𝑠C𝑝𝑞 and LsS𝑝𝑞 ∩ S, L𝑠𝐶𝑝𝑞 ∩ S, 0 q≤p≤10 𝑞≤𝑝≤1, defined by post quantum operators, which map the open unit disc 𝐷 onto the domain bounded by the limaçon curve defined by ∂Ds:={u+iv∈C:(u−1)2+v2−s4.2=4s2(u−1+s2)2+v2.}, where s∈−1,1.∖{0}. Keywords: Limaçon domain, subordination, (p, q)–derivative, Toeplitz and Hankel determinants, symmetric Toeplitz determinant, logarithmic coefficients, starlike functions with respect to symmetric points.

The purpose of this article is to obtain the sharp estimates of the first four initial logarithmic coefficients for the class BTs of bounded turning functions associated with a petal-shaped domain. Further, we investigate the sharp estimate of Fekete-Szegö inequality, Zalcman inequality on the logarithmic coefficients and the Hankel determinant H2,1Ff/2 and H2,2Ff/2 for the class BTs with the determinant entry of logarithmic coefficients.

Using the Lebedev–Milin inequalities, bounds on the logarithmic coefficients of an analytic function can be transferred to estimates on coefficients of the function itself and related functions. From this fact, the study of logarithmic-related problems of a certain subclass of univalent functions has attracted much attention in recent years. In our present investigation, a subclass of starlike functions Se* connected with the exponential mapping was considered. The main purpose of this article is to obtain the sharp estimates of the second Hankel determinant with the logarithmic coefficient as entry for this class.

In this paper, we introduce the subclass of star-like functions with respect to symmetric conjugate points associated with the sine function. Some coefficient functionals for this class are considered. Bounds of Taylor coefficients, logarithmic coefficients, and the Hankel and Toeplitz determinants whose entries are logarithmic coefficients are provided.

In recent years, many subclasses of univalent functions, directly or not directly related to the exponential functions, have been introduced and studied. In this paper, we consider the class of Se∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}^{\\ast}_{e}$\\end{document} for which zf′(z)/f(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$zf^{\\prime}(z)/f(z)$\\end{document} is subordinate to ez\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$e^{z}$\\end{document} in the open unit disk. The classic concept of Hankel determinant is generalized by replacing the inverse logarithmic coefficient of functions belonging to certain subclasses of univalent functions. In particular, we obtain the best possible bounds for the second Hankel determinant of logarithmic coefficients of inverse starlike functions subordinated to exponential functions. This work may inspire to pay more attention to the coefficient properties with respect to the inverse functions of various classes of univalent functions.

The aim of the present paper is to study some coefficient problems for certain classes associated with starlike functions such as sharp bounds for initial coefficients, logarithmic coefficients, Hankel determinants and Fekete–Szego problems. Moreover, we obtain some geometric properties as applications of differential subordinations.

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 Sμ,δ,qn,m and subordination are used in this study to define and construct new classes of α-convex functions associated with the cardioid domain. Additionally, this paper explores acute inequality problems for newly defined classes Rqα(a,c,m,L,P), of α-convex functions in the open unit disc Us, 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.

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.

Toeplitz and Hankel determinants of logarithmic coefficients for r-valent q-starlike and r-valent q-convex functions.

The purpose of this study was to obtain the sharp Hankel determinant H2,1Ff/2 and H2,2Ff/2 with a logarithmic coefficient as entry for the class BT3L of bounded turning functions connected with a three-leaf-shaped domain. In this study, we developed a novel method to prove the bound sharpness. Although the calculations are much easier using numerical analysis, all the proofs of our results can be checked with a basic knowledge of calculus.