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

Bednarz, U.; Kanas, S.

Artykuł - szczegóły

Tytuł artykułu

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

Autorzy

Bednarz U. , Kanas S.

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Warianty tytułu

Języki publikacji

Abstrakty

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.

Słowa kluczowe

k-uniformly convex functions k-starlike functions Hadamard product (or convolution) integral convolution integral transformation neighbourhoods of functions stability of convolution

k-jednostajne funkcje wypukłe k-funkcje gwieździste iloczyn Hadamarda splot całkowy całkowe przekształcenie otoczenie funkcji stabilność splotu

Wydawca

De Gruyter

Czasopismo

Journal of Applied Analysis

Rocznik

2004

Tom

Vol. 10, nr 1

Strony

105--115

Opis fizyczny

Bibliogr. 16 poz.

Twórcy

autor

Bednarz U.

Department of Mathematics Technical University of Rzeszów W. Pola 2 Pl-35-959 Rzeszów, Poland

autor

Kanas S.

Department of Mathematics Technical University of Rzeszów W. Pola 2 Pl-35-959 Rzeszów, Poland

Bibliografia

[1] Bednarz, U., Kanas, S. Generalized neighbourhoods and stability of convolution for the class of k-uniformly convex and k-starlike functions, Folia Sci. Univ. Tech. Resov. 175, Ser. Math. 23 (1999), 29-38.
[2] Bshouty, D., A note on Hadamard products of univalent functions, Proc. Amer. Math. Soc. 80 (1980), 271-272.
[3] Hayman, W. K., On the coefficients of univalent functions, Proc. Cambridge Philos. Soc. 55 (1959), 373-374.
[4] Kanas, S., Wiśniowska, A., Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105 (1999), 327-336.
[5] Kanas, S., Wiśniowska, A., Conic regions and k-uniform convexity II, Folia Sci. Univ. Tech. Resov. 170 (1998), 65-78.
[6] Kanas, S., Wiśniowska, A., Conic regions and k-starlike functions, Rev. Roumaine Math. Pures Appl. 45(3) (2000), 647-657.
[7] Kanas, S., Stability of convolution and dual sets for the class of k-uniformly convex and k-starlike functions, Folia Sci. Univ. Tech. Resov. 170 (1998), 51-64.
[8] Nezhmetdinov, I. Ft., Stability of geometric properties of convolutions of univalent functions, Russian Math. (Iz. VUZ) 37(11) (1993), 27-34.
[9] Pólya, G., Schoenberg, I. J., Remarks on de la Vallée-Poussin means and convex conformal maps of the circle, Pacific J. Math. 8 (1958), 295-334.
[10] Rahman, Q. I., Stankiewicz, J., On the Hadamard products of schlicht functions, Math. Nachr. 106 (1982), 7-16.
[11] Ruscheweyh, S., Duality for Hadamard product with applications to extremal problems, Trans. Amer. Math. 210 (1975), 63-74.
[12] Ruscheweyh, S., Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981), 521-527.
[13] Ruscheweyh, S., Sheil-Small, T., Hadamard product of schlicht functions and the Pólya-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119-135.
[14] Robertson, M. S., Applications of a lemma of Fejér to typically real functions, Proc. Amer. Math. Soc. 1 (1950), 555-561.
[15] Stankiewicz, J., Neighbourhoods of meromorphic functions and Hadamard products, Ann. Polon. Math. 46 (1985), 317-331.
[16] Stankiewicz, J., Stankiewicz, Z., Some classes of regular functions defined by convolution, Analytic functions, Błażejewko 1982 (Błażejewko, 1982), Lecture Notes in Math. 1039 (1983), Springer, Berlin, 400-408.

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Bibliografia

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