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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.

MSC:

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

Citations:

Zbl 1072.30009
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References:

[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 · doi:10.1155/S0161171204108090
[2] Sălăgean, GŞ, Subclasses of univalent functions, No. 1013, 362-372 (1983), Berlin, Germany · Zbl 0531.30009 · doi:10.1007/BFb0066543
[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
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