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Criteria of univalence for a certain integral operator

100%
Ignaciuk S., Parol M.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
|
2019
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tom 73
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nr 1
EN
In this article we consider the problem of univalence of a function introduced by Breaz and Ularu, improve some of their results and receive not only univalence conditions but also close-to-convex conditions for this function. To this aim, we used our method based on Kaplan classes.
2
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On solutions of integral equations with analytic kernels and rotations

100%
Nguyen M., Nguyen T.
Annales Polonici Mathematici
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1996
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tom 63
|
nr 3
293-300
EN
We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*)     x(t) + a(t)(Tx)(t) = b(t), where $T = M_{n₁,k₁} ... M_{n_m,k_m}$ and $M_{n_j,k_j}$ are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial $P_T(t) = t³ - t$. By means of the Riemann boundary value problems, we give an algebraic method to obtain all solutions of equation (*) in closed form.
3
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On some shift invariant integral operators, univariate case

63%
Anastassiou G., Gonska H.
Annales Polonici Mathematici
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1995
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tom 61
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nr 3
225-243
EN
In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is ℝ. A very general positive linear integral type operator is introduced through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and preservation of continuous probabilistic functions. Finally, four examples of very general specialized operators are presented fulfilling all the above properties; in particular, the inequalities for global smoothness preservation are proven to be sharp.
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