On certain new subclass of close-to-convex functions

In the present paper, we introduce a new subclass of the class of close-to-convex functions. The subordination and inclusion relationship, and some coefficient inequalities, a sufficient condition for this class are provided. Furthermore, we obtain the distortion theorems. The results presented here would provide extensions of those given in earlier works.     

On the properties of a certain subclass of close-to-convex functions

In the present paper, we introduce an interesting subclass Kps (h)of analytic functions in the open unit disk U. For functions belonging to theclass Kps (h), basic properties such as the coefficient bounds, the distortionand growth theorems are derived. The results presented here would provideextensions of those given by Q.-H. Xu et al. [2].     

Sharp distortion theorems for some subclass of close-to-convex functions

For real A, B such that −1 ? B < A ? 1, we denote by R(A,B) the class of functions, such that f′ ? (1 + Az)/(1 + Bz). Sharp distortion results for functions from R_A,B are obtained.     

Sharp distortion theorems for a subclass of close-to-convex mappings

We introduce the class of strongly close-to-convex mappings of order c~ in the unit ball of a complex Banach space, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space X or the unit polydisc in Cn. As an application, a sharp growth theorem for strongly close-to-convex mappings of order α is obtained.     

On a subclass of Bazilevic functions

A new subclass of Bazilevic functions is defined and some of its properties have been studied.     

On the successive coefficients of close-to-convex functions

Let f∈S, f be a close-to-convex function, f_k(z)=[f(z^k)]^1/k. The relative growth of successive coefficients of f_k(z) is investigated. The sharp estimate of ||c_n+1|−|c_n|| is obtained by using the method of the subordination function.     

Doubly close-to-convex functions

We introduce the class L(β,γ) of holomorphic, locally univalent functions in the unit disk , which we call the class of doubly close-to-convex functions. This notion unifies the earlier known extensions. The class L(β,γ) appears to be linear invariant. First of all we determine the region of variability for fixed z, |z|=r<1, which give us the exact rotation theorem. The rotation theorem and linear invariance allows us to find the sharp value for the radius of close-to-convexity and bound for the radius of univalence. Moreover, it was helpful as well in finding the sharp region for , for which the integral , f∈L(β,γ), is univalent. Because L(β,γ) reduces to β-close-to-convex functions (γ=0) and to convex functions (β=0 and γ=0), the obtained results generalize several corresponding ones for these classes. We improve as well the value of the radius of univalence for the class considered by Hengartner and Schober (Proc. Amer. Math. Soc. 28 (1971) 519-524) from 0.345 to 0.577.     

On k-uniformly close-to-convex functions of complex order

In this paper, we define and study some subclasses of analytic functions by using fractional derivative operator. Some interesting properties, coefficients problems and inclusion results of these classes are investigated. It is also shown that these classes are closed under convolution with convex functions and some applications are given.     

A generalization of a class of close-to-convex functions

We consider a class of functions, including the Bazilevich functions, which are regular in the unit disk. The theorems of Levandovskii identifying the class of close-to-convex functions with the class of linearly accessible functions are generalized, and the geometric structure of this class is established. A method constructing a subordinating homotopy chain is used.Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 509–516, May, 1972.     

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].