Coefficients Of Univalent Functions Research Articles

The Bound for Coefficients of Univalent Functions on the Superdisk

The Bound for Coefficients of Univalent Functions on the Superdisk

Sharp estimates of Newton coefficients of univalent functions

We consider the class of univalent holomorphic functions F(z) in the unit disk which are normalized by conditions F(0) = 0, F′(0) = 1. Estimates for the moduli of the Newton coefficients of these functions are established. It is shown that these estimates are sharp.

Smale’s mean value conjecture and the coefficients of univalent functions

We study Smale's mean value conjecture and its connection with the second coefficients of univalent functions. We improve the bound on Smale's constant given by Beardon, Minda and Ng.

On Keogh’s Length Estimate for Bounded Starlike Functions

For a bounded starlike function ƒ on the unit disc, we consider L(r), the length of the image of the circle ¦z¦ = r. Keogh showed that L(r) = O(log 1/(1 − r) as r → 1 and Hayman showed that this is the correct asymptotic. We give an alternative geometric construction which strengthens Hayman’s result, showing that the constant in Keogh’s original inequality is sharp. The analysis uses standard estimates on the hyperbolic metric of plane domains. The self-similarity of the construction allows for the examples to be expressed analytically. For context, we give a brief survey of related estimates on integral means and coefficients of univalent functions.

Conformally Invariant Functionals on the Riemann Sphere

The main aim of this work is to establish new inequalities for the Grunsky coefficients of univalent functions. For this purpose, we apply results from the theory of problems on extremal decomposition. To obtain inequalities for the Grunsky coefficients of a function $$f \in \Sigma$$ , we apply a solution of the problem on the maximum of a conformal invariant (this invariant, in its turn, is connected with the problem on extremal decomposition of $$\overline {\mathbb{C}}$$ into a family of simply connected and doubly connected domains). In contrast to similar inequalities obtained from the Jenkins general coefficient theorem, the inequalities established in this work are valid without any restrictions on the initial coefficients of the expansion of a function $$f \in \Sigma$$ . Bibliography: 6 titles.

Extremal results for a class of univalent functions

A variational method is used to obtain the sharp upper bound for the second coefficients of univalent functions which map onto regions of prescribed logarithmic capacity (Transfinite diameter).

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 Lowner about sharp bounds for coefficients of inverses of functions in the classS.

Estimates of logarithmic coefficients of univalent functions

In this paper, we derive some consequences of Milin's inequallty for the logarithmic coefficients of a univalent function by exploiting a reformulation of it.

On successive coefficients of univalent functions

SynopsisA coefficient difference bound for a class of univalent functions in the unit disk whose radial growth is large enough is established in this paper.

Coefficients of univalent functions of the Gel'fer class

Coefficients of univalent functions of the Gel'fer class

On Successive Coefficients of Univalent Functions

On Successive Coefficients of Univalent Functions

Mutual variation of the initial coefficients of univalent functions

Mutual variation of the initial coefficients of univalent functions

The Coefficients of Univalent Functions with Minigaps

The Coefficients of Univalent Functions with Minigaps

The Successive Coefficients of Univalent Functions

The Successive Coefficients of Univalent Functions

Successive coefficients of univalent functions

Successive coefficients of univalent functions

Estimates for inverse coefficients of univalent functions from integral means

The purpose of this note is to point out that sharp coefficient bounds for the inverses of univalent functions from certain families are fairly direct corollaries of results on integral means. As an example, in §1 the method is applied to the familiar schlicht classS. The resulting coefficient estimates for the inverses of functions inS were first obtained by K. Lowner. Following this prototype, in §2 we obtain corresponding results, which are new, for a classS(p) of meromorphic schlicht functions in |z|<1.

Goodman's Conjecture and the Coefficients of Univalent Functions

Goodman's Conjecture and the Coefficients of Univalent Functions

Goodman’s conjecture and the coefficients of univalent functions

Goodman’s conjecture (for a bound on the modulus of the nth coefficient of a p-valent function as a linear combination of the moduli of the first p coefficients) is considered in the special case of functions which are polynomials of univalent functions. For such functions, it is shown that Goodman’s conjecture is equivalent to a set of coefficient conjectures for normalized univalent functions.

On the local maxima of the coefficients of univalent functions

On the local maxima of the coefficients of univalent functions

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

For the coefficients bn 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 ¦bn¦≤1, if ¦b1¦≤0.524. It follows from (1) that the coefficients cn, 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, ¦cn¦≤n, if ¦c2¦≤1.046.