ON A SUBCLASS OF CLOSE TO CONVEX FUNCTIONS

Let C ′(α,β) be the class of functions f(z)=(a~nz~n)from n=2 to ∞ analytic in D ={z:|z|1},satisfying for some convex function g(z) with g(0) = g ′(0) ? 1 = 0 and for all z in D the condition ((zf′(z))/(g(z))-1)/(((zf′(z))/(g(z))+(1-2a))β for some α,β(0≤α1,0β≤1).A sharp coefficient estimate,distortion theorems and radius of convexity are determined for the class C ′(α,β).The results extend the work of C.Selvaraj.     

On some length problems for univalent functions

Let be the class of functions which are analytic in the unit disk . Let C(r) be the closed curve that is the image of the circle |z|=r < 1 under the mapping w = f(z), L(r) the length of C(r), and let A(r) be the area enclosed by the curve C(r). In 1968 D. K. Thomas shown that if , f is starlike with respect to the origin, and for 0≤r < 1, A(r) < A, an absolute constant, then Later, in 1969 Nunokawa has shown that if f is convex univalent, then This paper is devoted to obtaining a related correspondence between f(z) and L(r) for the case when f is univalent. Copyright © 2016 John Wiley & Sons, Ltd.     

A subclass of parabolic starlike and uniformly convex functions

Let A be the class of analytic functions in the open unit disk U. A function f in A satisfying the normalization is said to be in the class SP_n if Dⁿf is a parabolic starlike function, where Dⁿ is a notation of the Salagean operator. In this paper, several basic properties and characteristics of the class SP_n are investigated. These include subordination, convolution properties, class-preserving integral operators, and Fekete-Szegö problems.     

Strong starlikeness for a class of convex functions

By means of the Briot-Bouquet differential subordination, we estimate the order of strong starlikeness of strongly convex functions of a prescribed order. We also make numerical experiments to examine our estimates.     

Circular symmetrization,subordination and arclength problems on convex functions

We study the class of univalent analytic functions f in the unit disk of the form satisfying where Ω will be a proper subdomain of which is starlike with respect to . Let be the unique conformal mapping of onto Ω with and and . Let denote the arclength of the image of the circle , . The first result in this paper is an inequality for , which solves the general extremal problem , and contains many other well‐known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class .     

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.     

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.     

On a subclass of certain k-starlike functions with negative coefficients

The aim of the present paper is to show some properties of functions belonging to the class (k, n, α)− . The obtained results extend the results by Silverman [3].     

On uniformly convex functions

Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key to obtain other remarkable properties such as the coercivity. Our techniques allow to retrieve Enflo's uniformly convex renorming of super-reflexive Banach spaces as the regularization of a raw function built from trees. Among other applications, we provide a sharp estimation of the distance of a given function to the set of differences of Lipschitz convex functions. Finally, we prove the equivalence of several possible ways to quantify the super weakly noncompactness of a convex subset of a Banach space.     

On extremal problems related to integral transforms of a class of analytic functions

For γ?0 and β<1 given, let Pγ(β) denote the class of all analytic functions f in the unit disk with the normalization f(0)=f^′(0)−1=0 and satisfying the condition

     

Strongly hyperbolically convex functions

Let C(w₁,w₂,w₃) denote the circle in through w₁,w₂,w₃ and let denote one of the two arcs between w₁,w₂ belonging to C(w₁,w₂,w₃). We prove that a domain Ω in the Riemann sphere, with no antipodal points, is spherically convex if and only if for any w₁,w₂,w₃∈Ω, with w₁≠w₂, the arc of the circle which does not contain lies in Ω. Based on this characterization we call a domain G in the unit disk D, strongly hyperbolically convex if for any w₁,w₂,w₃∈G, with w₁≠w₂, the arc in D of the circle is also contained in G. A number of results on conformal maps onto strongly hyperbolically convex domains are obtained.     

On meromorphically harmonic starlike functions with respect to symmetric conjugate points

In a previous paper M.P. Chen, Z.-R. Wu and Z.-Z. Zou [M.P. Chen, Z.-R. Wu, Z.-Z. Zou, On functions α-starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 201 (1996) 25-34] developed a method, using some operators, to deal with functions analytic and starlike with respect to symmetric conjugate points in the unit disc. Then, the same method is employed to functions meromorphic by Z.Z. Zou and Z.-R. Wu [Zhong Zhu Zou, Zhuo-Ren Wu, On meromorphically starlike functions and functions meromorphically starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 261 (2001) 17-27]. Now, the method can be employed to functions meromorphic harmonic in the punctured disc 0<|z|<1. Especially, a sharp coefficient estimate and a structural representation of such functions are obtained.     

On approximately convex functions

A real valued function defined on a real interval is called -convex if it satisfies

The main results of the paper offer various characterizations for -convexity. One of the main results states that is -convex for some positive and if and only if can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case , the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called -convexity.

     

Closure properties of operators on the Ma-Minda type starlike and convex functions

A normalized univalent function f is called Ma-Minda starlike or convex if zf^′(z)/f(z)?φ(z) or 1+zf^″(z)/f^′(z)?φ(z) where φ is a convex univalent function with φ(0)=1. The class of Ma-Minda convex functions is shown to be closed under certain operators that are generalizations of previously studied operators. Analogous inclusion results are also obtained for subclasses of starlike and close-to-convex functions. Connections with various earlier works are made.     

Regions of variability for convex functions

Let ?? be the class of convex univalent functions f in the unit disc ?? normalized by f (0) = f ′(0) – 1 = 0. For z ₀ ∈ ?? and |λ | ≤ 1 we shall determine explicitly the regions of variability {log f ′(z ₀): f ∈ ??, f ″(0) = 2λ }. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Support-type properties of generalized convex functions

Chebyshev systems induce in a natural way a concept of convexity. The functions convex in this sense behave in many aspects similarly to ordinary convex functions. In this paper support-type properties are investigated. Using osculatory interpolation, the existence of support-like functions is established for functions convex with respect to Chebyshev systems. Unique supports are determined. A characterization of the generalized convexity via support properties is presented.     

On the continuity of biconjugate convex functions

We show that a Banach space is a Grothendieck space if and only if every continuous convex function on has a continuous biconjugate function on , thus also answering a question raised by S. Simons. Related characterizations and examples are given.