On certain general integral operators of analytic functions

B. A. Frasin

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2012)

  • Volume: 66, Issue: 1
  • ISSN: 0365-1029

Abstract

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In this paper, we obtain new sufficient conditions for the operators F α 1 , α 2 , . . . , α n , β ( z ) and G α 1 , α 2 , . . . , α n , β ( z ) to be univalent in the open unit disc 𝒰 , where the functions f 1 , f 2 , . . . , f n belong to the classes S * ( a , b ) and 𝒦 ( a , b ) . The order of convexity for the operators  F α 1 , α 2 , . . . , α n , β ( z ) and G α 1 , α 2 , . . . , α n , β ( z ) is also determined. Furthermore, and for β = 1 , we obtain sufficient conditions for the operators F n ( z ) and G n ( z ) to be in the class 𝒦 ( a , b ) . Several corollaries and consequences of the main results are also considered.

How to cite

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B. A. Frasin. "On certain general integral operators of analytic functions." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 66.1 (2012): null. <http://eudml.org/doc/289831>.

@article{B2012,
abstract = {In this paper, we obtain new sufficient conditions for the operators $F_\{\alpha _1,\alpha _2,...,\alpha _n,\beta \}(z)$ and $G_\{\alpha _1,\alpha _2,...,\alpha _n,\beta \}(z)$ to be univalent in the open unit disc $\mathcal \{U\}$, where the functions $f_1, f_2,..., f_n$ belong to the classes $S^*(a, b)$ and $\mathcal \{K\}(a, b)$. The order of convexity for the operators $F_\{\alpha _1,\alpha _2,...,\alpha _n,\beta \}(z)$ and $G_\{\alpha _1,\alpha _2,...,\alpha _n,\beta \}(z)$ is also determined. Furthermore, and for $\beta = 1$, we obtain sufficient conditions for the operators $F_n(z)$ and $G_n(z)$ to be in the class $\mathcal \{K\}(a, b)$. Several corollaries and consequences of the main results are also considered.},
author = {B. A. Frasin},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Analytic functions; starlike and convex functions; integral operator},
language = {eng},
number = {1},
pages = {null},
title = {On certain general integral operators of analytic functions},
url = {http://eudml.org/doc/289831},
volume = {66},
year = {2012},
}

TY - JOUR
AU - B. A. Frasin
TI - On certain general integral operators of analytic functions
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2012
VL - 66
IS - 1
SP - null
AB - In this paper, we obtain new sufficient conditions for the operators $F_{\alpha _1,\alpha _2,...,\alpha _n,\beta }(z)$ and $G_{\alpha _1,\alpha _2,...,\alpha _n,\beta }(z)$ to be univalent in the open unit disc $\mathcal {U}$, where the functions $f_1, f_2,..., f_n$ belong to the classes $S^*(a, b)$ and $\mathcal {K}(a, b)$. The order of convexity for the operators $F_{\alpha _1,\alpha _2,...,\alpha _n,\beta }(z)$ and $G_{\alpha _1,\alpha _2,...,\alpha _n,\beta }(z)$ is also determined. Furthermore, and for $\beta = 1$, we obtain sufficient conditions for the operators $F_n(z)$ and $G_n(z)$ to be in the class $\mathcal {K}(a, b)$. Several corollaries and consequences of the main results are also considered.
LA - eng
KW - Analytic functions; starlike and convex functions; integral operator
UR - http://eudml.org/doc/289831
ER -

References

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  11. Frasin. B. A., Aouf, M. K., Univalence conditions for a new general integral operator, Hacet. J. Math. Stat. 39 (2010), no. 4, 567-575. 
  12. Jabkubowski, Z. J., On the coefficients of starlike functions of some classes, Ann. Polon. Math. 26 (1972), 305-313. 
  13. Pascu, N., An improvement of Becker’s univalence criterion, Proceedings of the Commemorative Session: Simion Stoılow (Brasov, 1987), 43-48, Univ. Brasov, Brasov, 1987. 
  14. Pescar, V., A new generalization of Ahlfor’s and Becker’s criterion of univalence, Bull. Malaysian Math. Soc. (2) 19 (1996), no. 2, 53-54. 
  15. Seenivasagan, N., Sufficient conditions for univalence, Applied Math. E-Notes, 8 (2008), 30-35. 
  16. Seenivasagan, N., Breaz, D., Certain sufficient conditions for univalence, Gen. Math. 15 (2007), no. 4, 7-15. 

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