Carleson Measure of Harmonic Schwarzian Derivatives Associated with a Finitely Generated Fuchsian Group of the Second Kind

Let S H f be the Schwarzian derivative of a univalent harmonic function f in the unit disk D , compatible with a finitely generated Fuchsian group G of the second kind. We show that if S H f 2 1 − z 2 3 d x d y satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G , then S H f 2 1 − z 2 3 d x d y is a Carleson measure in D .

Some Characterizations of Weighted Holomorphic Function Classes by Univalent Function Classes

Some characterizations of Q K , ω p , q − type classes of holomorphic functions by Schwarzian derivatives with known conformal-type mappings are introduced in the present manuscript. Moreover, the action of the pre-Schwarzian derivatives on Q K , ω p , q − type classes, typically the univalent ones by using concerned Carleson-type measures, is investigated. In addition, we reveal important characterizations of some concerned weighted analytic-type spaces with the known Schwarzian derivatives evolving certain Q − type of concerned function class for a high utility toward practical and feasible application of concerned domains.

Estimate for Schwarzian derivative of certain close-to-convex functions

<abstract><p>Let $ f(z) $ be analytic in the unit disk with $ f(0) = f'(0)-1 = 0 $. For the following close-to-convex subclasses: $ \Re \{(1-z)f'(z)\} &gt; 0, $ $ \Re \{(1-z^{2})f'(z)\} &gt; 0, $ $ \Re \{(1-z+z^{2})f'(z)\} &gt; 0 $ and $ \Re \{(1-z)^{2}f'(z)\} &gt; 0 $, we investigate the bounds for the first two consecutive derivatives of higher order Schwarzian derivatives of $ f(z) $.</p></abstract>

Sharp coefficient bounds for starlike functions associated with the 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.

Hausdorff Dimension of Escaping Sets of Nevanlinna Functions

We determine the exact values of Hausdorff dimensions of escaping sets of meromorphic functions with polynomial Schwarzian derivatives. This will follow from the relation between these functions and the second-order differential equations in the complex plane.