Journal of Applied Analysis 10 (2004), No. 1, 105--115
Copyright Heldermann Verlag 2004
Stability of the Integral Convolution of k-Uniformly Convex and k-Starlike Functions
Urszula Bednarz
Dept. of Mathematics, Technical University, W. Pola 2, 35-959 Rzeszów, Poland
ubednarz@prz.edu.pl
Stanislawa Kanas
Dept. of Mathematics, Technical University, W. Pola 2, 35-959 Rzeszów, Poland
skanas@prz.edu.pl
[Abstract-pdf]
For a constant $k \in [0,\infty)$ 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 $k$-$\mathcal{UCV}$ [cf. S. Kanas, A. Wisniowska: Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105 (1999) 327--336]. The function $g$ is said to be $k$-starlike if $g(z) = zf'(z)$ and $f \in k$-$\mathcal{UCV}$. \par For analytic functions $f, g$, where $f(z) = z + a_2 z^2 + \cdots$ and $g(z) = z + \linebreak b_2 z^2 + \cdots$, the integral convolution is defined as follows: $$(f \otimes g)(z) =z+ \sum_{n=2}^\infty\frac{a_n b_n}{n}z^n .$$ \par In this note a problem of stability of the integral convolution of $k$-uniformly convex and $k$-starlike functions is investigated.
Keywords: k-uniformly convex functions, k-starlike functions, Hadamard product (or convolution), integral convolution, integral transformation, neighbourhoods of functions, stability of convolution.
MSC: 30C45; 30C50, 30C55
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