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Multidimensional analogs of the Fekete--Szeg\"{o} functional
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Abstract
In this paper we introduce the Fekete--Szeg\"{o} type mapping in the open unit ball of a complex Banach space. % and study its geometric and analytical properties. All previously studied modifications of the Fekete--Szeg\"{o} functional are either special cases or `components' of the mapping we introduce. The study involves the examination of transforms of the Fekete--Szeg\"o mapping under specific transformations applied to given holomorphic mappings. We show that for a mapping f, the third order Fr\'eshet derivative of the inverse mapping and of elements of the semigroup generated by f can be expressed in terms of the Fekete--Szeg\"{o} mapping. Estimates of the Fekete--Szeg\"o mapping over some subclasses of semigroup generators and of starlike mappings are also presented.
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In the first part of this paper, we will give the Fekete–Szegö inequality for various subfamilies of spirallike mappings of type β on the unit ball of a complex Banach space. Our results give extensions of those given by Lai and Xu (Results Math 76(4), Paper No. 191, 2021) and Elin and Jacobzon (Results Math 77(3), Paper No. 137, 2022). We next give the Fekete–Szegö inequality for close-to-quasi-convex mappings of type B on the unit ball of a complex Banach space. Our results give extensions of that given by Xu et al. (Complex Var Elliptic Equ. https://doi.org/10.1080/17476933.2021.1975115).
In the first part of this paper, we give generalizations of the Fekete–Szegö inequalities for quasiconvex mappings F of type B and the first elements F of g-Loewner chains on the unit ball of a complex Banach space, recently obtained by H. Hamada, G. Kohr and M. Kohr. We obtain the Fekete–Szegö inequalities using the norm under the restrictions on the second and third order terms of the homogeneous polynomial expansions of the mappings F. In the second part of this paper, we give the estimation of the difference of the moduli of successive coefficients for the first elements of g-Loewner chains on the unit disc. We also give the estimation of the difference of the moduli of successive coefficients for the first elements F of g-Loewner chains on the unit ball of a complex Banach space under the restrictions on the second and third order terms of the homogeneous polynomial expansions of the mappings F.
In this note we present a remark on the paper “On the coefficient inequalities for a class of holomorphic mappings associated with spirallike mappings in several complex variables” by Y. Lai and Q. Xu [10] published recently in the journal Results in Mathematics. We show that one of the theorems in [10] concerning the finite-dimensional space Cn is a direct consequence of another one, so it does not need an independent proof. Moreover, we prove that a sharp norm estimate on the Fekete–Szegö functional over spirallike mappings in a general Banach space can be deduced from a result in [10].
In this paper, we establish the Fekete and Szegö inequality for a class of holomorphic functions related to the class of normalized spirallike functions in the unit disk, and then we extend this result to higher dimensions. The results presented here would provide extensions of those given by Xu et al. (Sci China Math 61:1775–1788, 2018).
In this paper, we first give a coefficient inequality for holomorphic functions on the unit disc in which are subordinate to a holomorphic function p on with . Next, as applications of this theorem, we will give the Fekete-Szegö inequality for subclasses of normalized starlike mappings and normalized quasi-convex mappings of type B on the unit ball of a complex Banach space. We also give the Fekete-Szegö inequality for , where is the nonlinear resolvent of a mapping f in the Carathéodory family . Various particular cases will be also considered.
In the paper we discuss two functionals of the Fekete-Szegö type. πf(μ) = a2a4 - μa3² and ⊙f (μ) = a4 - μa²a³ for an analytic function f(z) = z + a2z² + a3z³ + ..., z ∈ Δ, (Δ = (z ∈ C. |z| < 1)) and a real number μ. We focus our research on the estimation of |πf (μ)| and |⊙;f (μ)|, while f is either in S* (the class of starlike functions) or in K (the class of convex functions).
We obtain sharp estimates for a generalized Zalcman coefficient functional with a complex parameter for the Hurwitz class and the Noshiro-Warschawski class of univalent functions as well as for the closed convex hulls of the convex and starlike functions by using an inequality from [5]. In particular, we generalize an inequality proved by Ma for starlike functions and answer a question from his paper [14]. Finally, we prove an asymptotic version of the generalized Zalcman conjecture for univalent functions and discuss various related or equivalent statements which may shed further light on the problem.
Let be the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known result for . We investigate the corresponding problem for the class of starlike mappings defined on the unit ball in a complex Banach space or on the unit polydisk in , which satisfies a certain condition.
Let S be the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known result maxf ∈ S|a3 - λ a2 2| = 1 + 2e-2λ/(1 - λ) for λ ∈ [ 0, 1 ]. We consider the corresponding problem for the family C of close-to-convex functions and get As an application it is shown that for close-to-convex functions, in contrast to the result in S $\max_{f \in S}| |a_3| - |a_2|| = 1.029\ldots.
In this paper, we will give the Fekete-Szegö inequality for the mappings f in various subclasses of normalized univalent mappings which are the first elements of g-Loewner chains on the unit disc U in C and also on the unit ball B of a complex Banach space. As an application, we give the estimation of the third coefficient for f under the condition that the second coefficient of f is zero. This result gives a generalization of the estimation of the third coefficient for odd univalent functions on the unit disc U. We also give the Fekete-Szegö inequality for the images of the first elements of g-Loewner chains on U under the Roper-Suffridge type extension operators.
"We study the FeketeSzego problem on the open unit ball of a complex Banach space. Namely, the FeketeSzego inequalities are proved for the class of spirallike mappings relative to an arbitrary strongly accretive operator, and some of its subclasses. Next, we consider families of non-linear resolvents for holomorphically accretive mappings vanishing at the origin. We solve the Fekete- Szego problem over these families."
Let C be the familiar class of normalized close-to-convex functions in the unit disk. In Koepf [On the Fekete-Szegö problem for close-to-convex functions. Proc Amer Math Soc. 1987;101:89–95], Koepf proved that for a function f(z)=z+∑k=2∞akzk in the class C, |a3−λa22|≤{3−4λ,λ∈[0,13],13+49λ,λ∈[13,23],1,λ∈[23,1]. As an important application, in the same paper, Koepf showed that ||a3|−|a2||≤1 for close-to-convex functions. In this paper, we extend the above results to a subclass of close-to-quasi-convex mappings of type B defined on the unit polydisc in Cn, and establish the sharp difference bound for the second and third coefficients of homogeneous expansions for this class of holomorphic mappings. The results presented here would provide a new path for solving the Bieberbach conjectures in several complex variables.
This book describes recent developments as well as some classical results regarding holomorphic mappings. The book starts with a brief survey of the theory of semigroups of linear operators including the Hille-Yosida and the Lumer-Phillips theorems. The numerical range and the spectrum of closed densely defined linear operators are then discussed in more detail and an overview of ergodic theory is presented. The analytic extension of semigroups of linear operators is also discussed. The recent study of the numerical range of composition operators on the unit disk is mentioned. Then, the basic notions and facts in infinite dimensional holomorphy and hyperbolic geometry in Banach and Hilbert spaces are presented, L. A. Harris' theory of the numerical range of holomorphic mappings is generalized, and the main properties of the so-called quasi-dissipative mappings and their growth estimates are studied. In addition, geometric and quantitative analytic aspects of fixed point theory are discussed. A special chapter is devoted to applications of the numerical range to diverse geometric and analytic problems.
In this paper, we establish the Fekete and Szegö inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in Cn.
Let be the familiar class of strongly starlike functions of order in the unit disk. Xu et al. (Results Math 72:343–357, 2017) proved that for a function in the class , then In this paper, we investigate the corresponding problem for the subclass of strongly starlike mappings of order defined on the unit ball in a complex Banach space, on the unit polydisk in and the bounded starlike circular domain in , respectively.
Nonlinear semigroup theory is not only of intrinsic interest, but is also important in the study of evolution problems. In the last forty years, the generation theory of flows of holomorphic mappings has been of great interest in the theory of Markov stochastic branching processes, the theory of composition operators, control theory, and optimization. It transpires that the asymptotic behavior of solutions to evolution equations is applicable to the study of the geometry of certain domains in complex spaces. Readers are provided with a systematic overview of many results concerning both nonlinear semigroups in metric and Banach spaces and the fixed point theory of mappings, which are nonexpansive with respect to hyperbolic metrics (in particular, holomorphic self-mappings of domains in Banach spaces). The exposition is organized in a readable and intuitive manner, presenting basic functional and complex analysis as well as very recent developments.
An unifying approach to the estimates of Fekete–Szegö functional |a3-μa22| are presented. Sharp bounds for that functional are found. References to a functions related to conic sections are given.
We consider semigroups of holomorphic self-mappings on domains in Hilbert and Banach spaces, and then develop a new dynamical approach to the study of geometric properties of biholomor- phic mappings. We establish, for example, several flow invariance conditions and find parametric representations of semicomplete vector fields. In order to examine the asymptotic behavior of these semigroups, we use diverse tools such as hyperbolic metric theory and estimates of so- lutions of generalized differential equations. In addition, we introduce a new method involving admissible upper and lower bounds. Finally, we apply our dynamical approach to obtain seve- ral growth and covering theorems for star-like mappings on the open unit balls of Banach and Hilbert spaces.
For nonzero complex b let Fn(b) denote the class of normalized univalent functions f satisfying Re [1+(z(Dnf)′(z)/Dnf(z)−1)/b]>0 in the unit disk U, where Dnf denotes the Ruscheweyh derivative of f. Sharp bounds for the Fekete–Szegö functional |a3−μa22| are obtained.
In this article, we provide a new method solving the Fekete-Szegö problem for classes of close-to-convex functions defined in terms of subordination. As an example, we apply it to the class of strongly close-to-convex functions.
Some results of Fekete-Szegö type for Bavrin's families of holomorphic functions in C n
- R Lugosz
- P Liczberski
R. D lugosz and P. Liczberski, Some results of Fekete-Szegö type for Bavrin's families of holomorphic functions in C n, Ann. Mat. Pura Appl. 4 200 (2021), 1841-1857,
https://doi.org/10.1007/s10231-021-01094-6.
Fekete-Szegö problem for Bavrin's functions and close-to-starlike mappings in C n
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R. D lugosz and P. Liczberski, Fekete-Szegö problem for Bavrin's functions and close-to-starlike mappings in C n, Anal. Math. Phys., 12 (2022),
https://doi.org/10.1007/s13324-022-00714-5.
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M. Elin and F. Jacobzon, Estimates on some functionals over non-linear resolvents,
Filomat, 37 (2023), 797-808.
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W. Rudin, Functional Analysis, Int. Ser. in Pure and Appl. Math. 8 NY: McGraw-Hill, 1991.
Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions
- T J Suffridge
T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Complex Analysis (Proc. Conf. Univ. Kentucky,
Lexington, KY, 1976), Lecture Notes in Math. 599 (1977), 146-159.