Radius of Convexity for Some Integral Operators on Function Spaces

  • Parvaneh NajmadiEmail author
  • Shahram Najafzadeh
  • Ali Ebadian
Research Paper
Part of the following topical collections:
  1. Mathematics


In this paper, we study the radius of convexity of the following integral operators
$$I_{n}^{{\gamma_{i} }} \left( {f_{1} , \ldots ,f_{n} } \right) = F\left( z \right) \, : = \mathop \int \limits_{0}^{z} \mathop \prod \limits_{i = 1}^{n} \left( {f_{i}^{'} \left( t \right)} \right)^{{\gamma_{i} }} {\text{d}}t$$
$$J_{n}^{{\gamma_{i} ,\lambda_{j} }} \left( {f_{1} , \ldots ,f_{n} ;g_{1} , \ldots ,g_{m} } \right) = J\left( z \right)\text{ := }\mathop \int \limits_{0}^{z} \mathop \prod \limits_{i = 1}^{n} \left( {f_{i}^{'} \left( t \right)} \right)^{{\gamma_{i} }} \mathop \prod \limits_{j = 1}^{m} \left( {\frac{{g_{j} \left( z \right)}}{z}} \right)^{{\lambda_{j} }} {\text{d}}t,$$
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.


Radii of convexity Starlike Convex Locally convex Integral operators Subordination Univalent 



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


  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, PrincetonGoogle Scholar
  2. Breaz D, Owa S, Breaz N (2008) A new integral univalent operator. Acta Univ Apulensis Math Inf 16:11–16MathSciNetzbMATHGoogle Scholar
  3. Dimkov G (1991) On products of starlike functions I. Ann Polon Math 55:75–79MathSciNetCrossRefGoogle Scholar
  4. Dimkov G, Dziok J (1998) Generalized problem of starlikeness for products of p-valent starlike functions. Serdica Math J 24:339–344MathSciNetzbMATHGoogle Scholar
  5. Ebadian A, Kargar R (2018) Univalence of integral operators on neighborhoods of analytic functions. Iran J Sci Technol Trans Sci 42:911–915MathSciNetCrossRefGoogle Scholar
  6. Frasin BA (2011) Order of convexity and univalency of general integral operator. J Frankl Inst 348:1013–1019MathSciNetCrossRefGoogle Scholar
  7. Kargar R, Pascu NR, Ebadian A (2017) Locally univalent approximations of analytic functions. J Math Anal Appl 453:1005–1021MathSciNetCrossRefGoogle 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 157–169Google Scholar
  9. Obradović M, Ponnusamy S, Wirths K-J (2013) Coefficient characterizations and sections for some univalent functions. Sib Math J 54:679–696MathSciNetCrossRefGoogle Scholar
  10. Ozaki S (1941) On the theory of multivalent functions II. Sci Rep Tokyo Bunrika Daigaku Sect A 4:45–87MathSciNetzbMATHGoogle 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 153–154Google Scholar
  12. Pescar V (1996) A new generalization of Ahlfor’s and Becker’s criterion of univalence. Bull Malays Math Soc (Second Ser) 19:53–54MathSciNetzbMATHGoogle Scholar
  13. Pommerenke C (1964) Linear-invariante familien analytischer funktionen. I. Math Ann 155:108–154MathSciNetCrossRefGoogle Scholar
  14. Silverman H (1975) Products of starlike and convex functions. Ann Univ Mariae Curie-Sk Lodowska 29:109–116MathSciNetzbMATHGoogle Scholar
  15. Silverman H (1999) Convex and starlike criteria. Int J Math Sci 22(1):75–79MathSciNetCrossRefGoogle Scholar
  16. Tuneski N (2003) On the quotient of the representations of convexity and starlikeness. Math Nachr 248–249:200–2003MathSciNetCrossRefGoogle Scholar

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© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of MathematicsPayame Noor UniversityTehranIran
  2. 2.Department of Mathematics, Faculty of SciencesUrmia UniversityUrmiaIran

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