Some radius problems related to a certain subclass of analytic functions

For real parameters α and β such that 0≤α1β,we denote by S(α,β) the class of normalized analytic functions which satisfy the following two-sided inequality:αR(zf′(z)/f(z))β,z∈U,where U denotes the open unit disk.We find a sufficient condition for functions to be in the class S(α,β) and solve several radius problems related to other well-known function classes.     

Univalence and starlikeness of certain transforms defined by convolution of analytic functions

Let U(λ) denote the class of all analytic functions f in the unit disk Δ of the form f(z)=z+a₂z²+? satisfying the condition

     

Integral means and Dirichlet integral for analytic functions

For normalized analytic functions f in the unit disk, the estimate of the integral means is important in certain problems in fluid dynamics, especially when the functions are non‐vanishing in the punctured unit disk . We consider the problem of finding the extremal function f which maximizes the integral means for f belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functional where denotes the area of the image of under . The first problem was originally discussed by Gromova and Vasil'ev in 2002 while the second by Yamashita in 1990.     

A study of a special family of analytic functions at infinity

In the present work we introduce certain new subclasses of analytic functions in the open unit disk U. Then by using a quasi-Hadamard product we discuss their applications in the space of analytic functions.     

On a class of analytic functions related to conic domains and associated with Carlson-Shaffer operator

Making use of the Carlson-Shaffer convolution operator, we introduce and study a new class of analytic functions related to conic domains. The main object of this paper is to investigat inclusion relations, coefficient bound for this class. We also show that this class is closed under convolution with a convex function. Some applications are also discussed.     

Certain subclasses of multivalent analytic functions defined by multiplier transforms

By making use of the principle of subordination between analytic functions and a family of multiplier transforms, we introduce and investigate some new subclasses of multivalent analytic functions. Such results as inclusion relationships, subordination and superordination properties, integral-preserving properties, argument estimates and convolution properties are proved.     

Differential subordinations concerning starlike functions

Denote byS ^* (⌕), (0≤⌕<1), the family consisting of functionsf(z)=z+a ₂z²+...+a_nzⁿ+... that are analytic and starlike of order ⌕, in the unit disc ⋎z⋎<1. In the present article among other things, with very simple conditions on μ, ⌕ andh(z) we prove the f’(z) (f(z)/z)^μ−1<h(z) implies f∈S^*(⌕). Our results in this direction then admit new applications in the study of univalent functions. In many cases these results considerably extend the earlier works of Miller and Mocanu [6] and others.     

Differential sobordination and Bazilevič functions

LetM(z)=z ⁿ +…,N(z)=z ⁿ +… be analytic in the unit disc Δ and let λ(z)=N(z)/zN′(z). The classical result of Sakaguchi-Libera shows that Re(M′(z)/N′(z))<0 implies Re(M(z)/N(z))>0 in Δ whenever Re(λ(z))>0 in Δ. This can be expressed in terms of differential subordination as follows: for anyp analytic in Δ, withp(0)=1,p(z)+λ(z)zp′(z)<1+z/1−z impliesp(z)<1+z/1−z, for Reλ(z)>0,z∈Δ. In this paper we determine different type of general conditions on λ(z),h(z) and ϕ(z) for which one hasp(z)+λ(z)zp′(z)<h(z) impliesp(z)<ϕ(z)<h(z) z∈Δ. Then we apply the above implication to obtain new theorems for some classes of normalized analytic funotions. In particular we give a sufficient condition for an analytic function to be starlike in Δ.     

On order of strongly starlikeness in the class of uniformly convex functions

In this paper we consider the order of strongly starlikeness in the classes uniformly convex functions.