Skip to main content

Radius of Convexity for Some Integral Operators on Function Spaces

Abstract

In this paper, we study the radius of convexity of the following integral operators

$$I_{n}^{{\gamma_{i} }} \left( {f_{1} , \ldots ,f_{n} } \right) = F\left( z \right) \, : = \mathop \int \limits_{0}^{z} \mathop \prod \limits_{i = 1}^{n} \left( {f_{i}^{'} \left( t \right)} \right)^{{\gamma_{i} }} {\text{d}}t$$

and

$$J_{n}^{{\gamma_{i} ,\lambda_{j} }} \left( {f_{1} , \ldots ,f_{n} ;g_{1} , \ldots ,g_{m} } \right) = J\left( z \right)\text{ := }\mathop \int \limits_{0}^{z} \mathop \prod \limits_{i = 1}^{n} \left( {f_{i}^{'} \left( t \right)} \right)^{{\gamma_{i} }} \mathop \prod \limits_{j = 1}^{m} \left( {\frac{{g_{j} \left( z \right)}}{z}} \right)^{{\lambda_{j} }} {\text{d}}t,$$

where \(f_{i} \left( {1 \le i \le n} \right)\) and \(g_{j } \left( {1 \le j \le m} \right)\) belong to some certain subclasses of analytic functions. Also, we investigate the univalency of

$$J_{n}^{{\gamma_{i} ,\lambda_{i} }} \left( {f_{1} , \ldots ,f_{n} ;\;g_{1} , \ldots ,g_{n} } \right)$$

under some conditions.

This is a preview of subscription content, access via your institution.

References

  1. Ahlfors LV (1974) Sufficient conditions for quasiconformal extension, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Ann Math Stud 79:23–29. Princeton Univ Press, Princeton

    Google Scholar 

  2. Breaz D, Owa S, Breaz N (2008) A new integral univalent operator. Acta Univ Apulensis Math Inf 16:11–16

    MathSciNet  MATH  Google Scholar 

  3. Dimkov G (1991) On products of starlike functions I. Ann Polon Math 55:75–79

    MathSciNet  Article  Google Scholar 

  4. Dimkov G, Dziok J (1998) Generalized problem of starlikeness for products of p-valent starlike functions. Serdica Math J 24:339–344

    MathSciNet  MATH  Google Scholar 

  5. Ebadian A, Kargar R (2018) Univalence of integral operators on neighborhoods of analytic functions. Iran J Sci Technol Trans Sci 42:911–915

    MathSciNet  Article  Google Scholar 

  6. Frasin BA (2011) Order of convexity and univalency of general integral operator. J Frankl Inst 348:1013–1019

    MathSciNet  Article  Google Scholar 

  7. Kargar R, Pascu NR, Ebadian A (2017) Locally univalent approximations of analytic functions. J Math Anal Appl 453:1005–1021

    MathSciNet  Article  Google Scholar 

  8. Ma WC, Minda D (1992) A unified treatment of some special classes of univalent functions. In: Proceedings of the conference on complex analysis. Int. Press, Cambridge, pp 157–169

  9. Obradović M, Ponnusamy S, Wirths K-J (2013) Coefficient characterizations and sections for some univalent functions. Sib Math J 54:679–696

    MathSciNet  Article  Google Scholar 

  10. Ozaki S (1941) On the theory of multivalent functions II. Sci Rep Tokyo Bunrika Daigaku Sect A 4:45–87

    MathSciNet  MATH  Google Scholar 

  11. Pascu N (1985), On a univalence criterion, II. In: Itinerant seminar on functional equations, approximation and convexity. Preprint85, Universitatea “Babes-Bolyai”, Cluj-Napoca, pp 153–154

  12. Pescar V (1996) A new generalization of Ahlfor’s and Becker’s criterion of univalence. Bull Malays Math Soc (Second Ser) 19:53–54

    MathSciNet  MATH  Google Scholar 

  13. Pommerenke C (1964) Linear-invariante familien analytischer funktionen. I. Math Ann 155:108–154

    MathSciNet  Article  Google Scholar 

  14. Silverman H (1975) Products of starlike and convex functions. Ann Univ Mariae Curie-Sk Lodowska 29:109–116

    MathSciNet  MATH  Google Scholar 

  15. Silverman H (1999) Convex and starlike criteria. Int J Math Sci 22(1):75–79

    MathSciNet  Article  Google Scholar 

  16. Tuneski N (2003) On the quotient of the representations of convexity and starlikeness. Math Nachr 248–249:200–2003

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This paper forms a part of Ph.D. thesis of the first author.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Parvaneh Najmadi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Najmadi, P., Najafzadeh, S. & Ebadian, A. Radius of Convexity for Some Integral Operators on Function Spaces. Iran J Sci Technol Trans Sci 43, 3029–3035 (2019). https://doi.org/10.1007/s40995-019-00791-5

Download citation

Keywords

  • Radii of convexity
  • Starlike
  • Convex
  • Locally convex
  • Integral operators
  • Subordination
  • Univalent