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1

A contribution on real and complex convexity in several complex variables

Abidi Jamel

Journal of Mathematics and Applications

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2022

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Vol. 45

21--58

EN

Let f, g : Cn → C be holomorphic functions. Define u(z, w) = |w − f (z)|4 + |w − g(z)|4, v(z, w) = |w − f (z)|2 + |w − g(z)|2, for (z, w) ∈ Cn × C. A comparison between the convexity of u and v is obtained under suitable conditions. Now consider four holomorphic functions φ1, φ2 : Cm → C and g1, g2 : Cn → C. We prove that F = |φ1 − g1|2 + |φ2 − g2|2 is strictly convex on Cn × Cm if and only if n = m = 1 and φ1, φ2, g1, g2 are affine functions with (φ′1g′2 − φ′2g′1)̸ = 0. Finally, it is shown that the product of four absolute values of pluriharmonic functions is plurisubharmonic if and only if the functions satisfy special conditions as well.

2

Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation

Khan Muhammad Bilal, Noor Muhammad Aslam, Macías-Díaz Jorge E., Soliman Mohamed S., Zaini Hatim Ghazi

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

387--403

EN

It is a well-known fact that inclusion and pseudo-order relations are two different concepts which are defined on the interval spaces, and we can define different types of convexities with the help of both relations. By means of pseudo-order relation, the present article deals with the new notions of convex functions which are known as left and right log- s -convex interval-valued functions (IVFs) in the second sense. The main motivation of this study is to present new inequalities for left and right log- s -convex-IVFs. Therefore, we establish some new Jensen-type, Hermite-Hadamard (HH)-type, and Hermite-Hadamard-Fejér (HH-Fejér)-type inequalities for this kind of IVF, which generalize some known results. To strengthen our main results, we provide nontrivial examples of left and right log- s -convex IVFs.

3

Radius problems for univalent functions

Sokół Janusz, Trąbka-Więcław Katarzyna

Journal of Applied Analysis

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2020

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Vol. 26, nr 1

111--115

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.

4

The real and complex convexity

Jamel A.

Journal of Mathematics and Applications

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2018

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Vol. 41

123--156

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.

5

The order of starlikeness of uniformly convex functions

Wiśniowska-Wajnryb A.

Journal of Applied Analysis

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2017

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Vol. 23, nr 1

33--38

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.

6

Certain subordination results on the convolution of analytic functions

Sharma P., Maurya R. K.

Journal of Mathematics and Applications

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2014

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Vol. 37

111--118

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].

7

Forecasting stock index movement direction with CPL linear classifier

Krawczuk J.

Zeszyty Naukowe Politechniki Białostockiej. Informatyka

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2011

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Z. 7

47-58

EN

Stocks, indexes, commodities, and precious metals price prediction is a difficult task where many approaches are used: traditional technical analysis, econometric time series or modern data mining techniques. One particular data mining technique - linear classifier is described in this article. Prediction based on linear classifier is done using current market state, which can be described by various data sets (attributes, features). The simplest form of this model could use data from yesterday’s price movement. Advanced models are using more historical price movements. Very advanced models include various historical price movements for indexes from other countries and other instruments like currencies, commodities, etc. Using more features requires extended time to estimate model parameters.We build the linear classifier models by the minimisation of a convex and piecewise-linear function which is very efficient comparing to other functions. Computational costs for building the model are similar to linear programming. We also use feature selection method called RLS. Those techniques allow us to explore data with many features. Four scenarios are considered, in each scenario a different amount of market data is used to create a model. In the simplest scenario only one day’s change in price is taken, in the most complicated one 421 historical prices of 43 different instruments are taken. Best results were achieved by using middle range of 52 attributes. In this scenario, the model was right 53.19% times. Meaning the directions of daily change in S&P500 index (up or down) were predicted correctly. This doesn’t seem a lot, but if those predictions would have been used for investing, they could produce a total profit of 77% in the tested time period from November 2008 to March 2011 (2 years 4 months), or an average of 28% per year.

PL

Prognozowanie cen akcji i wartośsci indeksów giełdowych jest zadaniem trudnym, dla którego użzywanych jest wiele różnych podejść. Artykuł ten przedstawia wprowadzenie do pewnych standardowych technik. Przedstawiona została tradycyjna analiza techniczna, ekonometryczne modele szeregów czasowych oraz współczesne metody eksploracji danych. Jedna z metod eksploracji danych, klasyfikator liniowy został przedstawiony bardziej szczegółowo. Został on użyty w przeprowadzonym eksperymencie do prognozowania wartości indeksu giełdy amerykańskiej. Prognozowanie takie oparte jest o dane opisujące obecny stan giełdy. Stan giełdy można opisać różną ilością danych (atrybutów, cech). W najprostszym przypadku może to być tylko jednodniowa zmiana ceny prognozowanego indeksu. W bardziej rozbudowanym modelu można użyć wielu cen historycznych. W modelu jeszcze bardziej rozbudowanym można użyć danych z innych giełd, kursów walut, cen towarów jak np. ropa. Użycie dużej ilości danych wymaga dłuższego czasu obliczeń parametrów modelu. W prezentowanym podejściu klasyfikator liniowy budowany jest w oparciu o minimalizację wypukłej i odcinkowo-liniowej funkcji kryterialnej. Metoda ta jest bardzo wydajna o koszcie zbliżonym do programowania liniowego. Dodatkowo użyta została metoda selekcji cech RLS. Techniki te pozwoliły na efektywną eksplorację danych o wielu wymiarach. W artykule przedstawiono cztery scenariusze o różnej ilości danych opisujących giełdę. W najprostszym użyto tylko jednej danej, w najbardziej rozbudowanym 421 danych o 43 instrumentach finansowych. Najlepsze wyniki uzyskano dla pośredniego modelu o 52 cechach, w którym model przewidział prawidłowo 53.19% kierunków dziennych zmian indeksu S&P500. Otrzymany wynik nie wydaje się być wysoki, jednak gdyby inwestowano w indeks zgodnie z modelem zysk z takich inwestycji wyniósłby 77% w okresie od października 2008 do marca 2011, dając średnio 28% zysku rocznie.

8

Properties of certain classes of analytic functions defined by Srivastava-Attiya operator

Khairnar S., More M.

Demonstratio Mathematica

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2010

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Vol. 43, nr 1

55-79

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.

9

Determination of convex bodies by infinity-chord functions

Sójka G.

Demonstratio Mathematica

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2010

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Vol. 43, nr 1

187-202

EN

In 1998 A. Soranzo introduced the notions of +infinity - and - infinity-chord functions (see [16]). In this paper we give an answer to the question when a convex body is determined by the values of -infinity-chord functions at chosen internal points. We also give some partial results regarding + infinity chord functions.

10

On classes of meromorphic or complex harmonic functions with a pole at the infinity

Jakubowski Z., Sibelska A.

Demonstratio Mathematica

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2009

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Vol. 42, nr 3

479-489

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].

11

Subclasses of typically-real functions defined by Ruscheweyh derivative

Lecko A., Lecko M., Yaguchi T.

Demonstratio Mathematica

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2008

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Vol. 41, nr 4

823-832

EN

For each lambda>- 1 let TR(lambda) be the class of all functions f analytic in D= {z is an element of C : \z\ < 1} of the form f (z) = [...] having real coefficients and satisfying the condition [...] where Llambda denote the Ruscheweyh derivative. Some basic properties of functions in are presented.

12

Some properties of a subclass of uniformly convex functions with negative coefficients

Aouf M., Mostafa A.

Demonstratio Mathematica

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2008

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Vol. 41, nr 2

353-370

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.

13

On quasi convex functions and Hadamard's inequality

Tseng K., Tseng K., Yang G., Dragomir S.

Demonstratio Mathematica

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2008

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Vol. 41, nr 2

323-336

EN

In this paper we establish some inequalities of Hadamard's type involving Godunova-Levin functions, P-functions, quasi-convex functions, J-quasi-convex functions, Wright-convex functions and Wright-quasi-convex functions.

14

Certain sufficienty conditions on Fox-Wright function

Chaurasia V., Parihar H.

Demonstratio Mathematica

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2008

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Vol. 41, nr 4

813-822

EN

The main object of this paper is to find certain conditions for the function [...] to be a member of certain subclasses of analytic functions. Our results provides generalization of some recent results due to Swaminathan [19] and Chaurasia and Srivastava [20].

15

Some relations including various linear operators

Dziok J.

Demonstratio Mathematica

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2007

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Vol. 40, nr 1

77-84

EN

Making use of the Carlson-Schaer linear operator, some subclasses of analytic functions are studied. Some relations including various linear operators are given.

16

On some new inequalities of Hermite-Hadamard-Fejer type involving convex functions

Tseng K., Hwang S., Dragomir S.

Demonstratio Mathematica

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2007

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Vol. 40, nr 1

51-64

EN

In this paper, we establish some inequalities of Hermite-Hadamard-Fejér type for m-convex functions and s-convex functions.

17

On certain subclasses of p-valently analytic functions of order alpha

Aouf M.

Demonstratio Mathematica

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2007

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Vol. 40, nr 2

317-330

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.

18

Functions convex in the positive direction of the imaginary axis

Lecko A.

Demonstratio Mathematica

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2007

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Vol. 40, nr 3

567-574

EN

The aim of this paper is to present a new method of the proof of an analytic characterization of functions convex in the positive direction of the imaginary axis.

19

A linear operator and associated class of multivalent analytic functions

Sokół J.

Demonstratio Mathematica

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2007

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Vol. 40, nr 3

559-566

EN

We introduce a certain class H alpha/k (p, lambda;h) of multivalent analytic functions in the open unit disc involving a linear operator L alpha/k. The aim of this paper is to extend the similar concept of many earlier papers. We use techniques of differential subordination and convolution of this class.

20

Special differential inequalities and their applications in the theory of regular function

Kowalczyk J.

Journal of Mathematics and Applications

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2006

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Vol. 28

85-91

EN

Let C denote the compIex pIane and Iet U denote the open unit disk. In this paper the second order nonlinear differential inequalities are investigated. Properties of the function which satisfies such differential inequalities are derived. Some applications in the theory of univalent function are presented.