Estimates for coefficients of univalent functions from integral means and Grunsky inequalities

Sharp bounds are obtained for the coefficients of inverses of univalent functions in the class Σ(p) by using results on integral means and generalized Grunsky inequalities. A new and elementary proof is given for a result due to Löwner about sharp bounds for coefficients of inverses of functions in the classS.     

The integral means spectrum of univalent functions

This is a survey of recent results on integral means of derivatives of univalent functions. Bibliography: 21 titles.Dedicated to the memory of Professor G. M. GoluzinTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 119–128.     

Estimates of the initial coefficients in the classes of univalent functions

Let S be the class of functions, regular and univalent in the circle ¦Z¦ < 1. Assume that D_n (), n=2,3,..., are defined by the expansion, ¦z¦<1,-11. In the paper one obtains sharp estimates for D₄() in the class S_R of functions from S with real coefficients C₂,C₃,... for all –1</1. In particular, as a consequence, one obtains sharp estimates for the coefficients C_3k+1 in the class S^K/R of K-symmetric functions from S_R for all K=2,3,... For k=2, the last result strengthens a result of Leeman.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 166–183, 1983.     

Estimates for the coefficients of univalent functions in terms of the second coefficient

For the coefficients b_n of an odd function \(f(z) = z + \sum\nolimits_{k = 1}^\infty {{}^bk^{z^{2k + 1} } } \) , regular in the unit disk, we obtain the estimate $$|b_n | \leqslant \frac{1}{{\sqrt 2 }}\sqrt {1 + |b_1 |^2 } \exp \frac{1}{2}\left( {\delta + \frac{1}{2}|b_1 |^2 } \right),where \delta = 0.312,$$ (1) from which it follows that ¦b_n¦≤1, if ¦b₁¦≤0.524. It follows from (1) that the coefficients c_n, n = 3, 4,..., of a regular function \(f(2) = z + \sum\nolimits_{k = 2}^\infty {{}^ck^{z^k } } \) , univalent in the unit desk, satisfy $$|c_n | \leqslant \frac{1}{2}\left( {1 + \frac{{|c_2 |^2 }}{4}} \right)n\exp \left( {\delta + \frac{{|c_2 |^2 }}{8}} \right),where \delta = 0.312,$$ in particular, ¦c_n¦≤n, if ¦c₂¦≤1.046.     

On successive coefficients of odd univalent functions

The relative growth of successive coefficients of odd univalent functions is investigated. We prove that a conjecture of Hayman is true.

     

Asymptotic estimates for the coefficients of bounded univalent functions

In the class of univalent bounded normalized holomorphic functions in the unit disk, we give an asymptotic estimate for the coefficients when the uniform norm of the modulus of the function tends to infinity.     

Concave univalent functions and Dirichlet finite integral

The article deals with the class consisting of non‐vanishing functions f that are analytic and univalent in such that the complement is a convex set, and the angle at ∞ is less than or equal to for some . Related to this class is the class of concave univalent mappings in , but this differs from with the standard normalization A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for settled by Avkhadiev et al. 3 . Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for is also presented.     

The sum of coefficients of bounded univalent functions

We solve the maximal value problem for the functional in the class of functionsf(z)=z+a ₂z²+… that are holomorphic and univalent in the unit disk and satisfy the inequality |f(z)|<M. We prove that the Pick functions are extremal for this problem for sufficiently largeM whenever the set of indicesk ₁,…,k_m contains an even number. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 728–733, May, 1997. Translated by S. S. Anisov     

Sub-Riemannian geometry of the coefficients of univalent functions

We consider coefficient bodies Mn for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then Mn are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system to calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case M₃.