Recently, many papers in the theory of univalent functions have been devoted to mapping and characterization properties of various linear integral or integro-differential operators in the class S (of normalized analytic and univalent functions in the open unit disk U), and in its subclasses (as the classes S∗ of the starlike functions and K of the convex functions in U). Among these operators, ...

A function is said to be bi-univalent on the open unit disk D if both the function and its inverse are univalent in D. Not much is known about the behavior of the classes of bi-univalent functions let alone about their coefficients. In this paper we use the Faber polynomial expansions to find coefficient estimates for four well-known classes of bi-univalent functions which are defined by subord...

Inequalities involving multipliers using the sequences {cn} and {dn} of positive real numbers are introduced for complex-valued harmonic univalent functions. By specializing {cn} and {dn}, we determine representation theorems, distortion bounds, convolutions, convex combinations, and neighbourhoods for such functions. The theorems presented, in many cases, confirm or generalize various well-kno...

We study the class of hyperbolically convex bounded univalent functions with a boundary normalization in the unit disk U . In the paper we obtain the lower estimate for the distortion in this class. A two-point distortion theorem is also proved. The method of proofs is based on the reduced modulus of digons and the modulus of annuli.

The logarithmic coefficients are very essential in the problems of univalent functions theory. importance is due to fact that bounds on f can transfer Taylor themselves or their powers, via Lebedev–Milin inequalities; therefore, it interesting investigate Hankel determinant whose entries coefficients. main purpose this paper obtain sharp for second strongly starlike and convex functions.

This is an introductory survey on applications of the Julia variation to problems in geometric function theory. A short exposition is given which develops a method for treating extremal problems over classes F of analytic functions on the unit disk D for which appropriate subsets Fn can be constructed so that (i) F = ⋃ n Fn and (ii) for each f ∈ Fn a geometric constraint will hold that ∂f(D) wi...

We say that a class F consisting of analytic functions f(z)=?n=0?anzn in the unit disk D:={z?C:|z|<1} satisfies Bohr phenomenon if there exists rf?(0,1) such that?n=1?|anzn|?d(f(0),?f(D)) for every function f?F and |z|=r?rf, where d is Euclidean distance. The largest radius rf F. In this paper, we establish classes Ma-Minda type starlike convex as well with respect to boundary point.

In this paper, we generalize and investigate the Bohr-Rogosinski inequalities property for subfamilies of univalent functions defined on unit disk D:={z∈C:|z|<1} which maps to concave domain, i.e., domain whose complement is a convex set. All results are proved be sharp.