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EN
This paper considers the following problem: for what value r, r < 1 a function that is univalent in the unit disk |z| < 1 and convex in the disk |z| < r becomes starlike in |z| < 1. The number r is called the radius of convexity sufficient for starlikeness in the class of univalent functions. Several related problems are also considered.
2
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.
3
Some inequalities for weighted harmonic and arithmetic operator means
EN
In this paper we establish some upper and lower bounds for the difference between the weighted arithmetic and harmonie operator means under various assumption for the positive invertible operators A, B. Some applications when A, B are bounded above and below by positive constants are given as well.
EN
We investigate classes of k-uniformly convex functions which generalize the class UCV introduced by Goodman in 1991. We solve the problem of finding the largest α ≥ 0 such that the class of k-uniformly convex functions is contained in the class of starlike functions of order α.
EN
In this paper, we derive general integral identity by establishing new Hermite-Hadamard type inequalities for functions whose absolute values of derivatives are convex and concave. Corresponding error estimates for midpoint formula are also included. Moreover, some applications to special means of real numbers are also provided.
6
Hermite-Hadamard type inequalities with applications
EN
In this article first, we give an integral identity and prove some Hermite-Hadamard type inequalities for the function ƒ such that |ƒ''|q is convex or concave for q ≥ 1. Second, by using these results, we present applications to ƒ-divergence measures. At the end, we obtain some bounds for special means of real numbers and new error estimates for the trapezoidal formula.
EN
We present Hermite-Hadamard type inequalities for Wright-convex, strongly convex and strongly Wright-convex functions of several variables defined on simplices.
EN
We define two new general integral operators for certain analytic functions in the unit disc U and give some sufficient conditions for these integral operators on some subclasses of analytic functions.
9
Inequalities of Jensen type for φ-convex functions
EN
Some inequalities of Jensen type for φ-convex functions defined on real intervals are given.
10
Certain subordination results on the convolution of analytic functions
EN
In this paper, certain subordination results on the convolution of finite number of analytic functions are derived. Our results include a sufficiency condition for convexity of the convolution of analytic functions fi satisfying [wzór].
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.
12
EN
In this article we investigate some classes of meromorphic or complex harmonie functions with a pole, which are generated either by analytic conditions or by "coefficient inequalities". There are given theorems, which combine the geometrical properties of functions of the introduced classes. Some results broaden knowledge about the classes of functions, which were investigated in [15]. The main inspiration for the reaserch were the papers [4] and [11]. The part of results were presented in the XII-th International Mathematically-Informatical Conference in Chełm (2nd-5th July, 2006) [12].
13
Remarks on the certain subclass of univalent functions
EN
We investigate the family LP α (α ∈ (-π, π]) of functions [wzór] that are analytic in the unit disk with the property that the domain of values [wzór] is the parabolic region (Imw) ² < 2Rew - 1.We give inclusion theorems and bounds of Re �'(z) for this class.
14
On certain properties of neighborhoods of analytic functions of complex order
EN
Let A(n) denote the class of functions of the form [wzór] which are analytic in the open unit disk U = {z : |z| < 1}. In this note, the subclasses Sn (β, γ, a, c), Rn (β, γ, a, c; μ ), S(sup α) (sub n) (β, γ a, c) and R (sup α) (sub n) (β, γ, a, c; μ ) of A(n)(are defined and some properties of neighborhoods arę studied for functions of complex order in these classes.
15
On an application of certain sufficient condition for starlikeness
EN
In this paper we consider a sufficient condition for furiction to be a-starlike function, when α ∈ [0, 1/2]. We use it for certain subclass of strongly starlike functions defined by a geometric condition. We take advantage of the techniąues of differential subordinations.
16
Classes of functions defined by subordination
EN
In the paper, we define classes of analytic functions, in terms of subordination. We present some inclusion relations for defined classes.
17
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.
18
On a subclass of uniformly convex functions with fixed second coefficient
EN
Using of Salagean operator, we define a new subclass of uniformly convex functions with negative coefficients and with fixed second coefficient. The main objective of this paper is to obtain coefficient estimates, distortion bounds, closure theorems and extreme points for functions belonging of this new class. The results are generalized to families with fixed finitely many coefficients.
19
The convexity of Hadamard product of three functions
EN
Let α, β, γ < and let f, g, h be analytic functions such that Re[f'(z)] ≥ α, Re[g'(z)] ≥ β, Re[h'(z)] ≥ γ in the unit disc U. In this paper we give a sufficient condition for the convexity of Hadamard product f *g *h in the unit dsc U.
20
Some parametric family of functions
EN
In the paper, we define classes of analytic functions, in terms of some linear operator. Coefficients estimates, distortion theorems, and the radii of convexity and starlikeness for this class of analytic functions are given here.
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