A new general integral operator defined by Al-Oboudi differential operator. (English) Zbl 1166.30006

The author defines a new integral operator as follows: Let \(k\in \mathbb{N}_0\), \(l=(l_{1},\dots ,l_{n})\in \mathbb{N}_{0}^{n}\), and \(\mu_{i}>0\, ,\, 1\leq i\leq n\, .\) One defines an integral operator \(I_{k,n,l,\mu}: A^{n}\rightarrow A\) by \[ I_{k,n,l,\mu}(f_{1},\dots ,f_{n})=F, \]
\[ \displaystyle{D^{k}F(z)= \int_{0}^{z} \left (\frac{D^{l_{1}}f_{1}(t)}{t}\right )^{\mu_{1}} \dots \left ( \frac{D^{l_{n}}f_{n}(t)}{t}\right )^{\mu_{n}} dt}\, , \]
where \(f_{1}\, ,\, \dots\, ,\, f_{n}\in A\) and \(D\) is the Al-Oboudi differential operator. Also he introduces new subclasses of analytic functions and gives some results concerning the integral operator \(I_{k,n,l,\mu}\) on the above mentioned subclasses. The results presented on this paper generalize some results of M. Acu, D. Breaz, H.O. Guney and Gr. Salagean.


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
47G10 Integral operators


Zbl 1072.30009
Full Text: DOI EuDML


[1] Al-Oboudi FM: On univalent functions defined by a generalized Sălăgean operator.International Journal of Mathematics and Mathematical Sciences 2004,2004(27):1429-1436. 10.1155/S0161171204108090 · Zbl 1072.30009
[2] Sălăgean, GŞ, Subclasses of univalent functions, No. 1013, 362-372 (1983), Berlin, Germany · Zbl 0531.30009
[3] Frasin BA: Family of analytic functions of complex order.Acta Mathematica. Academiae Paedagogicae Nyíregyháziensis 2006,22(2):179-191. · Zbl 1120.30302
[4] Magdaş I: , Doctoral thesis. University “Babeş-Bolyai”, Cluj-Napoca, Romania; 1999.
[5] Acu M: Subclasses of convex functions associated with some hyperbola.Acta Universitatis Apulensis 2006, (12):3-12. · Zbl 1164.30320
[6] Breaz D, Breaz N: Two integral operators.Studia Universitatis Babeş-Bolyai. Mathematica 2002,47(3):13-19. · Zbl 1027.30018
[7] Breaz D, Güney HÖ, Sălăgean GŞ: A new general integral operator.Tamsui Oxford Journal of Mathematical Sciences. Accepted Tamsui Oxford Journal of Mathematical Sciences. Accepted · Zbl 1200.30006
[8] Bulut, S., Some properties for an integral operator defined by Al-Oboudi differential operator (2008) · Zbl 1149.30015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.