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1
Content available remote On q-analogue of Janowski-type starlike functions with respect to symmetric points
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.
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.
3
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.
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.
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Content available remote On generalization of close-to-convexity for complex holomorphic functions in Cn
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]).
6
Content available remote Holomorphic extension of locally holder functions
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.
7
Content available remote A proof of the Ehrenpreis-Martineau theorem using the Bochner-Martinelli kernel
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.
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 [...].
EN
In this paper a sufficient univalence criterion is given (Theorem 1). This result is an essential generalization of the well-known univalence criterions.
EN
This is an expository note, cocncerning the properties of exceptional sets for functions from the Bergman space in domains in Cx.
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).
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.
13
Content available remote Barycentric transformations
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.
14
Content available remote On the Łojasiewicz exponent of the gradient of holomorphic functions
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].
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