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EN
In the class of analytic functions in the unit disc |z| < 1 we prove some new sufficient conditions for functions to be univalent or to be close-to-convex in the unit disc. Also we extend Ozaki’s condition that Re{exp(iα)f (p)(z)} > 0 in |z| < 1 implies that f(z) is at most p-valent in |z| < 1.
EN
We give negative answer to the question of Bordulyak and Sheremeta for more general classes of entire functions than in the original formulation: Does index boundedness in joint variables for an entire function F imply index boundedness in the variable zj for the function F? This question is addressed for entire functions of bounded L-index in joint variables and entire functions of bounded L-index in direction. We also present a class of analytic functions in the unit ball which has bounded L-index in joint variablesand has unbounded l-index in the variables z1 and z2 for any positive continuous function l : B2→C.
3
The real and complex convexity
EN
We prove that the holomorphic differential equation ϕ’’(ϕ+c) = γ(ϕ’)² (ϕ:C→C be a holomorphic function and (γ, c) ϵ C²) plays a classical role on many problems of real and complex convexity. The condition exactly γ ϵ [wzór] (independently of the constant c) is of great importance in this paper. On the other hand, let n ≥ 1, (A₁, A₂) ϵ C² and g₁, g₂ : Cᵑ → C be two analytic functions. Put u(z, w) = │A ₁w - g₁(z) │² + │A₂w - g₂(z) │²v(z,w) = │A₁w - g₁(z) │² + │ A₂w - g₂(z) │², for (z,w) ϵ Cᵑ x C. We prove that u is strictly plurisubharmonic and convex on Cᵑ x C if and only if n = 1, (A₁, A₂) ϵ C² \{0} and the functions g₁ and g₂ have a classical representation form described in the present paper. Now v is convex and strictly psh on Cᵑ x C if and only if (A₁, A₂) ϵ C² \{0}, n ϵ {1,2} and and g₁, g₂ have several representations investigated in this paper.
4
Cr-right equivalence of analytic functions
EN
Let f, g : (Rn , 0) → (R, 0) be analytic functions. We will show that if (…) and (…) then f and g are Cr-right equivalent, where (f) denoteideal generated by f and (…).
5
A class of univalent functions involving a differentio-integral operator
EN
This paper focuses on a generalized linear operator Im which is a combination of both differential and integral operators. Involving this operator, a class Tsk(...) with respect to k-symmetric points is defined. Results based on coefficient inequalities and bounds for this class are obtained. Various integral representations and some consequent results for TS(...) class are also determined. Further, results on partial sums are discussed.
EN
In this paper we introduce the class K(s,b,beta,apha) of analytic functions defined by the Srivastava-Attiya convolution operator Js,b(f) involving the Hurwitz-Lerch Zeta function. We derive few subordination results for the functions in the class K(s,b,beta,alpha) and discuss the interesting applications of subordination results with the help of convex functions. Several other properties like coefficient inequalities growth and distortion theorems, extreme points, integral mean inequalities, partial sums and quasi-Hadamard product are investigated for the class K(s,b,beta,alpha). The authors also obtain Fekete-Szego inequality for normalized analytic functions f(z) defined on the open unit disc for which [....] lies in a region starlike with respect to 1 and is symmetric with respect to the real axis. Applications of our main result involving the Owa-Srivastava operator of fractional calculus are discussed. Finally as one of the applications of our result, we derive the Fekete-Szego inequality for a class of normalized analytic functions, defined using the Hadamard product and the Owa-Srivastava operator.
EN
Using the Wright's generalized hypergeometric function, we introduce a new class Wk (p, q, s; A, B, lambda) of analytic p-valent functions with negative coefficients. In this paper we investigate coefficients estimates, distortion theorem, the radii of p-valent starlikeness and p-valent convexity and modified Hadamard products.
EN
In this paper, we obtain some applications of first order differential subordination and superordination results involving Dziok-Srivastava operator and other linear operators for certain normalized analytic functions in the open unit disk. The results, which are presented in this paper, have relevant connections with various previous results.
EN
The aim of this paper is to obtain coefficient estimates, distortion theorem, extreme points and radii of close - to - convexity, starlikeness and convexity for functions belonging to the subclass TSlambda (n,alpha, beta) of uniformly convex functions with negative coefficients. We also derive many results for the modified Hadamard products of functions belonging to the class TSlambda(n,alpha, beta), and obtain several interesting distortion theorems for certain fractional operators of functions in this class. Finally, we consider integral operators associated with functions in this class.
10
On partial sums of certain analytic functions
EN
In the present paper we give some results concerning partial sums of certain analytic functions analogous to the results due to H. Silverman [J. Math. Anal. Appl. 209 (1997), 221-227]. All the results are sharp.
11
On certain class of analytic functions associated with a convolution structure
EN
Making use of a convolution structure, we introduce a new class of analytic functions defined in the open unit disc and investigate its various characteristics. Apart from deriving a set of coefficient bounds, we establish several inclusion relation-ships involving the (n, [...)-neighborhoods of analytic functions with negative coefficients belonging to this subclass.
12
On a subclass of p-valent functions
EN
In this paper we consider a subclass of p-valent functions defined by certain differential-integral operator. By using the Krein-Milman theorem we obtain the extreme points of the classs. Some extremal problems in the class are also determined.
EN
By making use of the familiar concept of neighbourhood of analytic and p-valent functions, the author prove coefficient bounds and distortion inequalities and associated inclusion relations for the (j, [...)-neighbourhoods of a family of p-valent functions with negative coefficients and defined by using Salagean operator which is defined by means of a certain non-homogenous Cauchy-Euler differential equation.
EN
The authors establish certain results concerning the generalized Hadamard products of certain meromorphic univalent functions with positive coefficients analagous to the results due to Choi et al. (J. Math. Anal. Appl. 199(1996), 495-501).
15
Distortion theorems in the class Kn(D)
EN
This article presents the analysis of properties of functions belonging to the class Kn(D) earlier introduced by the author. This is a class of functions analytical in the domain D, for which the n-th divided difference [F; zo, . . . , zn] [...]consisting of functions univalent in the domain D is obtained. The subclass Kn(E), formed by functions F(z) = zn + an+izn+1 + .... analytical in a unit circle E is separated from class Kn(D) and its properties are considered. The author touches classical and at the same time urgent questions, arising at the study of analytical functions, belonging to some class.
16
On certain subclasses of p-valently analytic functions of order alpha
EN
The object of the present paper is to derive various properties and char- acteristics of certain subclasses of p-valently analytic functions of order alpha in the open unit disc by using the techniques involving the Briot-Bouquet differential subordination.
17
Certain subclasses of multivalent prestarlike functions with negative coefficients
EN
The object of the present paper is to investigate coefficient estimates and closure theorems for functions belonging to the class R[..] of p-valent prestarlike functions with negative coefficients. We also consider integral operators associalted with functions belonging to the class R[..]. Furthermore, distortion theorems involving a generalized fractional integral (derivative) opertaor for functions in this class are proved.
18
Quasihomographies of the real line II
EN
In paper [2] we can find growth theorems for quasihomographies of the real line with three poiots (x1, x2, ∞) fixed. Those estimatioos were given for x > x1. ln this paper we give the estimations for x < x1.
19
On sufficient condition for starlikeness of certain integral of analytic function
EN
The author aims at finding certain condition on Re f' which is sufficient for the function Φ(z) = ∫(sup z)(sub 0) (sup f(t))(sub t)dt to be starlike function.
20
Radius problems for a class of analytic functions
EN
In this present paper, we investigate certain radius problems for a class of normalized analytic functions in the open unit disk in the complex plane.
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