On the coefficients of a bounded univalent analytic function

LetF _M denote the class of univalent analytic functionsf in |z|<1 with the expansionf (z)=z+a ₂ z ²+a ₃ z ³+... and |f(z)|?M in |z|<1. In this note I derive a rough bound for alln-th coefficients and a more accurate bound for all the third coefficients of functionsf belonging toF _M.     

The Schwarzian derivative for harmonic mappings

Journal d'Analyse Mathématique -     

Lorentzian worldlines and the Schwarzian derivative

The aim of this note is to relate the classical Schwarzian derivative and the geometry of Lorentz surfaces of constant curvature. Université de la Méditerranée and CPT-CNRS, CNRS, Centre de Physique Théorique, Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 69–72, April–June, 2000. Translated by C. Duval and V. Ovsienko     

Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities

We prove that if is of bounded variation, then the uncentered maximal function is absolutely continuous, and its derivative satisfies the sharp inequality . This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.

     

Locally convex functions and the Schwarzian derivative

In this paper we study various classes of locally convex analytic functions in the unit disc, which are invariant under the group of M?bius automorphisms of the unit disc. Bounds for the Schwarzian derivative of functions in these classes are achieved and used to obtain estimates for the uniform hyperbolic radius of univalence in these classes.     

The Schwarzian derivative and the Wiman-Valiron property

Let f be a transcendental meromorphic function in the plane such that f has finitely many critical values, the multiple points of f have bounded multiplicities, and the inverse function of f has finitely many transcendental singularities. Using the Wiman-Valiron method, it is shown that if the Schwarzian derivative S_f of f is transcendental, then f has infinitely many multiple points, the inverse function of S_f does not have a direct transcendental singularity over ∞, and ∞ is not a Borel exceptional value of S_f. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method.     

About the rate function in concentration inequalities for suprema of bounded empirical processes

We provide new deviation inequalities in the large deviations bandwidth for suprema of empirical processes indexed by classes of uniformly bounded functions associated with independent and identically distributed random variables. The improvements we get concern the rate function which is, as expected, the Legendre transform of the suprema of the log-Laplace transform of the pushforward measure by the functions of the considered class (up to an additional corrective term). Our approach is based on a decomposition in martingale together with some comparison inequalities.     

Some inequalities for the polar derivative of a polynomial

LetP(z) be a polynomial of degreen which does not vanish in |z|<1. In this paper, we estimate the maximum and minimum moduli of thekth polar derivative ofP(z) on |z|=1 and thereby obtain compact generalizations of some known results, which among other results, yields interesting refinements of Erdos-Lax theorem and a theorem of Ankeny and Rivlin.     

A generalization of the Schwarzian derivative and its applications

A generalization of the Schwarzian derivative is proposed; this generalization is used to investigate functions having divided differences which are nonzero in a given region. A relation is established between this class of functions and linear homogeneous differential equations.Translated from Matematicheskie Zametki, Vol. 10, No. 2, pp. 229–238, August, 1971.     

Interpolating sequences for bounded analytic functions

We prove that any sequence in the open ball of a complex Banach space even in that of whose norms are an interpolating sequence for is interpolating for the space of all bounded analytic functions on The construction made yields that the interpolating functions depend linearly on the interpolated values.

     

Some integral inequalities for the polar derivative of a polynomial

If P(z) is a polynomial of degree n which does not vanish in |z| 1,then it is recently proved by Rather [Jour.Ineq.Pure and Appl.Math.,9 (2008),Issue 4,Art.103] that for every γ 0 and every real or complex number α with |α|≥ 1,{∫02π |D α P(e iθ)| γ dθ}1/γ≤ n(|α| + 1)C γ{∫02π|P(eiθ)| γ dθ}1/γ,C γ ={1/2π∫0 2π|1+eiβ|γdβ}-1/γ,where D α P(z) denotes the polar derivative of P(z) with respect to α.In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J.Approx.Theory,54 (1988),306-313] as a special case.     

关于有界函数导数的估计

主要讨论了有界函数的导数估计问题,得到三阶导数、四阶导数的准确估计式。     

Dichotomy results for delay differential equations with negative Schwarzian derivative

We gain further insight into the use of the Schwarzian derivative to obtain new results for a family of functional differential equations including the famous Wright’s equation and the Mackey-Glass type delay differential equations. We present some dichotomy results, which allow us to get easily computable bounds of the global attractor. We also discuss related conjectures, and formulate new open problems.     

Dynamics of a family of transcendental meromorphic functions having rational Schwarzian derivative

In the present paper, a class F of critically finite transcendental meromorphic functions having rational Schwarzian derivative is introduced and the dynamics of functions in one parameter family is investigated. It is found that there exist two parameter values λ^∗=?(0)>0 and , where and is the real root of ?^′(x)=0, such that the Fatou sets of fλ(z) for λ=λ^∗ and λ=λ∗∗ contain parabolic domains. A computationally useful characterization of the Julia set of the function fλ(z) as the complement of the basin of attraction of an attracting real fixed point of fλ(z) is established and applied for the generation of the images of the Julia sets of fλ(z). Further, it is observed that the Julia set of fλ∈K explodes to whole complex plane for λ>λ∗∗. Finally, our results found in the present paper are compared with the recent results on dynamics of one parameter families λtanz, [R.L. Devaney, L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. 22 (4) (1989) 55-79; L. Keen, J. Kotus, Dynamics of the family λtan(z), Conform. Geom. Dynam. 1 (1997) 28-57; G.M. Stallard, The Hausdorff dimension of Julia sets of meromorphic functions, J. London Math. Soc. 49 (1994) 281-295] and , λ>0 [G.P. Kapoor, M. Guru Prem Prasad, Dynamics of : The Julia set and bifurcation, Ergodic Theory Dynam. Systems 18 (1998) 1363-1383].