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1

On Fibonacci numbers in edge coloured trees

Bednarz U., Bród D., Szynal-Liana A., Włoch I., Wołowiec-Musiał M.

Opuscula Mathematica

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2017

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Vol. 37, no. 4

479--490

EN

In this paper we show the applications of the Fibonacci numbers in edge coloured trees. We determine the second smallest number of all (A, 2B)-edge colourings in trees. We characterize the minimum tree achieving this second smallest value.

2

Distance Fibonacci numbers, their interpretations and matrix generators

Bednarz U., Włoch A., Wołowiec-Musiał M.

Commentationes Mathematicae

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2013

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Vol 53, No. 1

35--46

EN

In this paper we define a distance Fibonacci numbers, also for negative integers, which generalize the classical Fibonacci numbers and Padovan numbers, simultaneously. We give different interpretations of these numbers with respect to special partitions and compositions, also in graphs. We show a construction of the sequence of distance Fibonacci numbers using the Pascal’s triangle. Moreover, we give matrix generators of these numbers, for negative integers, too.

3

On T-neighborhoods of analytic functions

Bednarz U., Sokół J

Journal of Mathematics and Applications

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2010

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Vol. 32

25-32

4

Stability of the integral transformation of k-uniformly convex and k-starlike functions

Bednarz U.

Journal of Mathematics and Applications

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2007

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Vol. 29

91-98

EN

For a constant k ϵ [0, ∞) a normalized function f, analytic in the unit disk, is said to be k-uniformly convex if Re (1+z f" (z)/f'(z)) > k|zf"(z)/f'(z)| at any point in the unit disk. The class of k-uniformly convex functions is denoted k-UCV (cf. [8]). The function g is said to be k-starlike if g(z) = zf'(z) and f ϵ k-UCV. For analytic function f, where f(z) = z + a2z² + źźź the integral transformation is defined as follows: [wzór]. Generalized neighbourhood is defined as: [wzór]. In this note a problem of stability of the integral transformation of k-uniformly convex and k-starlike functions for TNδ neighbourhoods is investigated.

5

Stability of the Hadamard product of k-uniformly convex and k-starlike functions in certain neighbourhood

Bednarz U.

Demonstratio Mathematica

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2005

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Vol. 38, nr 4

837--845

6

Stability of the integral convolution of k-uniformly convex and k-starlike functions

Bednarz U., Kanas S.

Journal of Applied Analysis

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2004

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Vol. 10, nr 1

105-115

EN

For a constant k ∈ [0, ∞) a normalized function f, analytic in the unit disk, is said to be k-uniformly convex if Re(1 + zf"(z)/f'(z)) > k|zf"(z)/f'(z)| at any point in the unit disk. The class of k-uniformly convex functions is denoted k-UCV (cf. [4]). The function g is said to be k-starlike if g(z) = zf'(z) and f ∈ k-UCV. For analytic functions f, g, where f(z) = z + a2z² + • • • and g(z) = z + b2z² + • • •, the integral convolution is defined as follows: [wzór] In this note a problem of stability of the integral convolution of k-uniformly convex and k-starlike functions is investigated.

7

Generalized neighbourhoods and stability of convolution for the class of k-uniformly convex and k-starlike functions

Bednarz U., Kanas S.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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1999

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z. 23 [175]

29-38

EN

Let k-UCV denote the class of k-uniformly convex functions and let k-ST be the related class of k-starlike functions. For [..] > 0 and nonnegative sequence {tn] we define generalized neighbourhood [...].