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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

On q-analogue of Janowski-type starlike functions with respect to symmetric points

Khan Muhammad Ghaffar, Ahmad Bakhtiar, Khan Raees, Zubair Muhammad, Salleh Zabidin

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

37--46

EN

The main objective of the present paper is to define a class of q-starlike functions with respect to symmetric points in circular domain. Some interesting results of these functions have been evaluated in this article. The sufficiency criteria in the form of convolutions are evaluated. Furthermore, other geometric properties such as coefficient bounds, distortion theorem, closure theorem and extreme point theorem are also obtained for these newly defined functions.

3

An estimate for the distance of a complex valued Sobolev function defined on the unit disc to the class of holomorphic functions

Fuchs M.

Journal of Applied Analysis

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2011

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Vol. 17, nr 1

131-135

EN

Let f : B →C denote a Sobolev function of class W1p defined on the unit disc. We show that the distance of f to the class of all holomorphic functions measured in the norm of the space W1p(B;C) is bounded by the Lp-norm of theWirtinger derivative ∂-zf. As a consequence we obtain a Korn type inequality for vector fields B →R2.

4

New properties of some families of holomorphic functions of several complex variables

Leś-Bomba E., Liczberski P.

Demonstratio Mathematica

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2009

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Vol. 42, nr 3

491-503

EN

The paper concerns holomorphic functions in complete bounded n-circular domains of the space Cn and presents some properties of the above mentioned functions belonging to the families described by some geometrical or analytical conditions. This subject has been considered by many mathematicans, for example I.I. Bavrin, K. Dobrowolska, I. Dziubiński, S. Fukui, Z. J. Jakubowski, J. Kamiński, A. Marchlewska, Y. Michiwaki, J. A. Pfaltzgraff, R. Sitarski, T. J. Suffridge, J. Stankiewicz, I. Weinberg, A. Wrzesień, Ł. Żywień and the authors.

5

On some correspondence between holomorphic functions in the unit disc and holomorphic functions in the left halfplane

Ligocka E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2009

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Vol. 57, no 3-4

223-229

EN

We study a correspondence L between some classes of functions holomorphic in the unit disc and functions holomorphic in the left halfplane. This correspondence is such that for every ƒ and w ∈ H, exp(L(ƒ)(w)) = ƒ(exp w). In particular, we prove that the famous class S of univalent functions on the unit disc is homeomorphic via L to the class S(H) of all univalent functions g on H for which = g(w) + 2 πi) = g(w) + 2 πi and limRe z →-- ∞(g(w) - w)=0.

6

On generalization of close-to-convexity for complex holomorphic functions in Cn

Marchlewska A.

Demonstratio Mathematica

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2005

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Vol. 38, nr 4

847--856

EN

Sveral authors (I. I. Bawrin [1], K. Dobrowolska, I. Dziubinski, P. Liczberski, R. Sitarski [3], [4], [5], [13], S. Gong, S. S. Miller [6], Z. J. Jakubowski and J. Kaminski [8], J. Janiec [10] and others) studied various families of complex holomorphic functions in Cn and in Banach space, corresponding with famous subclasses of univalent functions. In this paper we study a class of holomorphic functions of n complex variables analogous to the class of close-to-convex functions of one variable considered by M. Biernacki, W. Kaplan and Z. Lewandowski (see [2], [11], [12]).

7

Holomorphic extension of locally holder functions

Rusev P.

Demonstratio Mathematica

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2003

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Vol. 36, nr 2

335--342

EN

Let γ be a smooth Jordan curve in the extended complex plane passing trough the point at infinity. In the paper are given sufficiently conditions under which a complex function defined on γ admits a holomorphic extensions into a region complementary to γ.

8

A proof of the Ehrenpreis-Martineau theorem using the Bochner-Martinelli kernel

Hatziafratis T.

Demonstratio Mathematica

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2003

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Vol. 36, nr 1

93--97

EN

We give an elementary proof of a version of the Ehrenpreis-Martineau theorem, which describes the entire holomorphic functions of exponential type as combinations of exponential functions. The proof uses the Bochner-Martinelli kernel and appropriate derivatives of it.

9

Relations betweensome subclasses of Caratheodory class

Jurasińska R.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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2002

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z. 26 [199]

77-87

EN

Let U = {z is an element of C : \z\ < 1} denote the unit disc and let H = H(U) denote the family of functions holomorphic in U. Let omega denote the class of Schwarz functions w is an element of H such that [...]. We say that / is subordinate to g in U and write [...].

10

On certain extension of univalence criterions for holomorphic functions in the unit disc

Bogowski F., Wesołowski A.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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2001

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z. 25 [190]

47-53

EN

In this paper a sufficient univalence criterion is given (Theorem 1). This result is an essential generalization of the well-known univalence criterions.

11

Exceptional sets for holomographic functions

Jakóbczak P.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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2001

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z. 25 [190]

63-70

EN

This is an expository note, cocncerning the properties of exceptional sets for functions from the Bergman space in domains in Cx.

12

On classes of functions defined by subordination to some special n-valent functions

Jurasińska R.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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1999

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z. 23 [175]

52-60

EN

Let H = H(U) be the class of all functions which are holomorphic in the unit disc U = {z : \z[ < 1}. Let P(n,A,B) denotes the class of all functions p(z) = 1 +p1z +p2z2 + ...is an element of H, such that p(z) -< 1+Azn/1-Bzn, where -< denotes subordination. With the class P(n, A, B) we connect the subclass S*(n, A, B) of starlike functions in the following way. A function f(z) = z o+a2z.2 z2 + ... belongs to S*(n, A, B) if and only ifzf'(z)/f(z) is an element of P(n, A, B). In this note we give some estimations for the modulus of functions and coefficients in the classes P(n,A,B) and S*(n,A, B).

13

Linear-invariant order for integral transform of univalent functions

Lewandowski Z., Prokhorov D., Szynal A.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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1999

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z. 23 [175]

71-74

EN

For alpha is an element of R, we find the linear-invariant order for functions g(z) = integralz/0 f'(t)(f(t)/t)alpha dt where f(z) = z+.... are holomorphic and univalent in the unit disk.

14

Barycentric transformations

Mazouzi S.

Demonstratio Mathematica

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1999

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Vol. 32, nr 2

287-296

EN

We present in this note a new family of automorphisms on spaces of holomorphic functions called Barycentric Transformations. Beside the theoretical aspect of these transformations we shall use them to solve explicitly barycentric differential equations of the form.

15

On the Łojasiewicz exponent of the gradient of holomorphic functions

Bogusławska M.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 4

337-343

EN

We show that the Łojasiewicz exponent L[sub o] (grad f) at zero of the gradient of a distinguished pseudo-polynomial f [...] is attained on the zero-set of f'[sub y].