Convolution conditions for bounded α -starlike functions of complex order

A. Y. Lashin

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2017)

  • Volume: 71, Issue: 1
  • ISSN: 0365-1029

Abstract

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Let A be the class of analytic functions in the unit disc U of the complex plane ℂ with the normalization f ( 0 ) = f ' ( 0 ) - 1 = 0 . We introduce a subclass S M * ( α , b ) of A , which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class S M * ( n , α , b ) ( n ≥ 0 ) related to S M * ( α , b ) is also considered under the same conditions. Among other things, we find convolution conditions for a function f ∈ A to belong to the class S M * ( α , b ) . Several properties of the class S M * ( n , α , b ) are investigated.

How to cite

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A. Y. Lashin. "Convolution conditions for bounded $\alpha $-starlike functions of complex order." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 71.1 (2017): null. <http://eudml.org/doc/289796>.

@article{A2017,
abstract = {Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $\mathbb \{C\}$ with the normalization $f(0)=f^\{^\{\prime \}\}(0)-1=0$. We introduce a subclass $S_\{M\}^\{\ast \}(\alpha ,b)$ of $A$, which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_\{M\}^\{\ast \}(n,\alpha ,b)$ ($n\ge 0$) related to $S_\{M\}^\{\ast \}(\alpha ,b)$ is also considered under the same conditions. Among other things, we find convolution conditions for a function $f\in A$ to belong to the class $S_\{M\}^\{\ast \}(\alpha ,b)$. Several properties of the class $S_\{M\}^\{\ast \}(n,\alpha ,b)$ are investigated.},
author = {A. Y. Lashin},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Univalent functions; bounded starlike functions of complex order; bounded convex functions of complex order; $\alpha $-starlike functions},
language = {eng},
number = {1},
pages = {null},
title = {Convolution conditions for bounded $\alpha $-starlike functions of complex order},
url = {http://eudml.org/doc/289796},
volume = {71},
year = {2017},
}

TY - JOUR
AU - A. Y. Lashin
TI - Convolution conditions for bounded $\alpha $-starlike functions of complex order
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2017
VL - 71
IS - 1
SP - null
AB - Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $\mathbb {C}$ with the normalization $f(0)=f^{^{\prime }}(0)-1=0$. We introduce a subclass $S_{M}^{\ast }(\alpha ,b)$ of $A$, which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_{M}^{\ast }(n,\alpha ,b)$ ($n\ge 0$) related to $S_{M}^{\ast }(\alpha ,b)$ is also considered under the same conditions. Among other things, we find convolution conditions for a function $f\in A$ to belong to the class $S_{M}^{\ast }(\alpha ,b)$. Several properties of the class $S_{M}^{\ast }(n,\alpha ,b)$ are investigated.
LA - eng
KW - Univalent functions; bounded starlike functions of complex order; bounded convex functions of complex order; $\alpha $-starlike functions
UR - http://eudml.org/doc/289796
ER -

References

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  1. Aouf, M. K., Darwish, H. E., Attiya, A. A., On a class of certain analytic functions of complex order, Indian J. Pure Appl. Math. 32 (10) (2001), 1443-1452. 
  2. El-Ashwah, R. M., Some convolution and inclusion properties for subclasses of bounded univalent functions of complex order, Thai J. Math. 12 (2) (2014), 373-384. 
  3. Goodman, A. W., Univalent Functions, Vol. I, II, Tampa, Florida, 1983. 
  4. Kulshrestha, P. K., Bounded Robertson functions, Rend. Mat. 6 (1) (1976), 137-150. 
  5. Kulshrestha, P. K., Distortion of spiral-like mapping, Proc. Royal Irish Acad. 73A (1973) 1-5. 
  6. Libera, R. J., Univalent λ -spiral functions, Canad. J. Mat. 19 (1967) 449-456. 
  7. Nasr, M. A., Aouf, M. K., On convex functions of complex order, Mansoura Sci. Bull. 9 (1982), 565-582. 
  8. Nasr, M. A., Aouf, M. K., Bounded convex functions of complex order, Mansoura Sci. Bull. 10 (1983), 513-526. 
  9. Nasr, M. A., Aouf, M. K., Bounded starlike functions of complex order, Proc. Indian Acad. Sci. (Math. Sci.) 92 (2) (1983) 97-102. 
  10. Nasr, M. A., Aouf, M. K., Starlike functions of complex order, J. Natur. Sci. Math. 25 (1985), 1-12. 
  11. Padmanabhan, K. S., Ganesan, M. S., Convolution conditions for certain class of analytic functions, Indian J. Pure Appl. Math. 15 (1984), 777-780. 
  12. Robertson, M. S., On the theory of univalent functions, Ann. of Math. 37 (1936), 374-408. 
  13. Salagean, G. S., Subclasses of univalent functions, in: Complex Analysis – Fifth Romanian-Finnish Seminar (C. A. Cazacu, N. Boboc, M. Jurchescu, I. Suciu, eds.) Springer, Berlin–Heidelberg, 1983, 362-372. 
  14. Silverman, H., Silvia, E. M., Telage, D., Convolution conditions for convexity and starlikeness and spiral-likeness, Math. Z. 162 (1978), 125-130. 
  15. Sizuk, P. I., Regular functions f ( z ) for which z f ' ( z ) is λ -spirallike, Proc. Amer. Math. Soc. 49 (1975), 151-160. 
  16. Singh, R., Singh, V., On a class of bounded starlike functions, Indian J. Pure Appl. Math. 5 (8) (1974), 733-754. 
  17. Wiatrowski, P., The coefficient of a certain family of holomorphic functions, Zeszyty Nauk. Uniw. Łódz. Nauki Mat. Przyrod. Ser. II No. 39 Mat. (1971), 75-85. 

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