Coefficient estimates for a class of meromorphic bi-univalent functions

Applying the Faber polynomial coefficient expansions to a class of meromorphic bi-univalent functions, we obtain the general coefficient estimates for such functions and also examine their early coefficient bounds. A function univalent in the open unit disk is said to be bi-univalent if its inverse map is also univalent there. Both the technique and the coefficient bounds presented here are new on their own kind. We hope that this article will generate future interest in applying our approach to other related problems.     

Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions

Estimates on the initial coefficients are obtained for normalized analytic functions f in the open unit disk with f and its inverse g=f⁻¹ satisfying the conditions that zf^′(z)/f(z) and zg^′(z)/g(z) are both subordinate to a univalent function whose range is symmetric with respect to the real axis. Several related classes of functions are also considered, and connections to earlier known results are made.     

Coefficient estimates for a certain class of analytic and bi-univalent functions defined by fractional derivative

We introduce and investigate a subclass of analytic and bi-univalent functions defined by a fractional derivative operator in the open unit disk. Using the Faber polynomial expansions, we obtain upper bounds for the coefficients of functions belonging to this class.     

Asymptotic estimates of certain integral means for meromorphic functions. II

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 6, pp. 844–847, June, 1989.     

On a certain classes of meromorphic functions with positive coefficients

In this paper certain classes of meromorphic functions in punctured unit disk are defined. Some properties including coefficient inequalities, convolution and other results are investigated.     

Some applications of Miller-Mocanu lemma on certain classes of meromorphic functions

In this paper, making use of a linear operator we introduce and study certain new classes of meromorphic functions. We derive some inclusion results. These classes contain many known classes as a special case.     

Norm estimates for convolution transforms of certain classes of analytic functions

For normalized analytic functions f in the unit disk Δ, we consider the class

     

Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions

In this work, considering a general subclass of analytic bi-univalent functions, we determine estimates for the general Taylor–Maclaurin coefficients of the functions in this class. For this purpose, we use the Faber polynomial expansions. In certain cases, our estimates improve some of those existing coefficient bounds.     

Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

For ?1≤B<A≤1, let \(\mathcal {S}^{*}(A,B)\) denote the class of normalized analytic functions \(f(z)= z+{\sum }_{n=2}^{\infty }a_{n} z^{n}\) in |z|<1 which satisfy the subordination relation z f ^′(z)/f(z)?(1 + A z)/(1 + B z) and Σ^?(A,B) be the corresponding class of meromorphic functions in |z|>1. For \(f\in \mathcal {S}^{*}(A,B)\) and λ>0, we shall estimate the absolute value of the Taylor coefficients a _n (?λ,f) of the analytic function (f(z)/z)^?λ . Using this we shall determine the coefficient estimate for inverses of functions in the classes \(\mathcal {S}^{*}(A,B)\) and Σ^?(A,B).     

Iteration of certain exponential-like meromorphic functions

The dynamics of functions \(f_\lambda (z)= \lambda \frac{\mathrm{e}^{z}}{z+1}\ \text{ for }\ z\in \mathbb {C}, \lambda >0\) is studied showing that there exists \(\lambda ^* > 0\) such that the Julia set of \(f_\lambda \) is disconnected for \(0< \lambda < \lambda ^*\) whereas it is the whole Riemann sphere for \(\lambda > \lambda ^*\). Further, for \(0< \lambda < \lambda ^*\), the Julia set is a disjoint union of two topologically and dynamically distinct completely invariant subsets, one of which is totally disconnected. The union of the escaping set and the backward orbit of \(\infty \) is shown to be disconnected for \(0<\lambda < \lambda ^*\) whereas it is connected for \(\lambda > \lambda ^*\). For complex \(\lambda \), it is proved that either all multiply connected Fatou components ultimately land on an attracting or parabolic domain containing the omitted value of the function or the Julia set is connected. In the latter case, the Fatou set can be empty or consists of Siegel disks. All these possibilities are shown to occur for suitable parameters. Meromorphic functions \(E_n(z) =\mathrm{e}^{z}(1+z+\frac{z^2}{2!}+\cdots +\frac{z^n}{n!})^{-1}\), which we call exponential-like, are studied as a generalization of \(f(z)=\frac{\mathrm{e}^{z}}{z+1}\) which is nothing but \(E_1(z)\). This name is justified by showing that \(E_n\) has an omitted value 0 and there are no other finite singular value. In fact, it is shown that there is only one singularity over 0 as well as over \(\infty \) and both are direct. Non-existence of Herman rings are proved for \(\lambda E_n \).     

Coefficient estimates for a certain family of analytic functions involving a q-derivative operator

The Ramanujan Journal - In this paper, we study the coefficient estimates for a class of analytic functions defined by using the q-Ruscheweyh derivative operator. In particular, we investigate the...