Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 719354, 4 pages doi:10.1155/2008/719354 Research Article Certain Integral Operators on the Classes Mβ i  and Nβ i  Daniel Breaz Department of Mathematics, 1st December 1918, University of Alba Iulia, 510009 Alba, Romania Correspondence should be addressed to Daniel Breaz, dbreaz@uab.ro Received 13 September 2007; Revised 21 October 2007; Accepted 2 January 2008 Recommended by Vijay Gupta We consider the classes Mβ i  and Nβ i  of the analytic functions and two general integral opera- tors. We prove some properties for these operators on these classes. Copyright q 2008 Daniel Breaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let U  {z ∈ C, |z| < 1} be the open unit disk and let A denote the class of the functions fz of the form fzz  a 2 z 2  a 3 z 3  ···,z∈ U, 1.1 which are analytic in the open disk U. Let Mβ be the subclass of A, consisting of the functions fz, which satisfy the inequal- ity Re  zf  z fz  <β, z∈ U,β>1, 1.2 and let Nβ be the subclass of A, consisting of functions fz, which satisfy the inequality Re  zf  z f  z  1  <β, z∈ U. 1.3 These classes are studied by Uralegaddi et al. in 1, and Owa and Srivastava in 2. Consider the integral operator F n introduced by D. Breaz and N. Breaz in 3,havingthe form F n z  z 0  f 1 t t  α 1 ···  f n t t  α n dt, 1.4 where f i z ∈Aand α i > 0, for all i ∈{1, ,n}. 2 Journal of Inequalities and Applications Remark 1.1. This operator extends the integral operator of Alexander given by Fz  z 0 ft/tdt. Also, we consider the next integral operator denoted by F α 1 , ,α n that was introduced by Breaz et al. in 4,havingtheform F α 1 , ,α n z  z 0  f  1 t  α 1 ···  f  n t  α n dt, 1.5 where f i z ∈Aand α i > 0 for all i ∈{1, ,n}. It is easy to see that these integral operators are analytic operators. 2. Main results Theorem 2.1. Let f i ∈Mβ i , for each i  1, 2, 3, ,n with β i > 1.ThenF n z ∈Nμ with μ  1   n i1 α i β i − 1 and α i > 0,(i  1, 2, 3, ,n). Proof. After some calculi, we obtain that zF  n z F  n z  n  i1 α i zf  i z f i z − n  i1 α i . 2.1 The relation 2.1 is equivalent to Re  zF  n z F  n z  1   n  i1 α i Re  zf  i z f i z  − n  i1 α i  1. 2.2 Since f i ∈Mβ i ,wehave Re  zF  n z F  n z  1  < n  i1 α i β i − n  i1 α i  1  n  i1 α i  β i − 1   1. 2.3 Because  n i1 α i β i − 1 > 0, we obtain that F n ∈Nμ,whereμ  1   n i1 α i β i − 1. Corollary 2.2. Let f i ∈Mβ for each i  1, 2, 3, ,n with β>1.ThenF n z ∈Nγ with γ  1 β − 1  n i1 α i and α i > 0, i  1, 2, 3, ,n. Proof. In Theorem 2.1, we consider β 1  β 2  ···  β n  β. Corollary 2.3. Let f ∈Mβ with β>1. Then the integral operator Fz  z 0 ft/t α dt ∈Nδ with δ  αβ − 11 and α>0. Proof. In Corollary 2.2, we consider n  1andα 1  α. Corollary 2.4. Let f ∈Mβ with β>1. Then the integral operator of Alexander Fz  z 0 ft/tdt ∈Nβ. Daniel Breaz 3 Proof. We have zF  z F  z  zf  z fz − 1. 2.4 From 2.4,wehave Re  zF  z F  z  1   Re zf  z fz <β. 2.5 So relation 2.5 implies that Alexander operator is in Nβ. Theorem 2.5. Let f i ∈Nβ i  for each i  1, 2, 3, ,n,withβ i > 1.ThenF α 1 , ,α n z ∈Nρ with ρ  1   n i1 α i β i − 1 and α i > 0, i  1, 2, 3, ,n. Proof. After some calculi, we have zF  α 1 , ,α n z F  α 1 , ,α n z  α 1 zf  1 z f  1 z  ···  α n zf  n z f  n z 2.6 that is equivalent to zF  α 1 , ,α n z F  α 1 , ,α n z  1  α 1  zf  1 z f  1 z  1   ···  α n  zf  n z f  n z  1  − n  i1 α i  1. 2.7 Since f i ∈Nβ i , for all i ∈{1, ,n},wehave Re  zf  n z f  n z  1  <β i . 2.8 So we obtain Re  zF  α 1 , ,α n z F  α 1 , ,α n z  1  < n  i1 α i β i − n  i1 α i  1  n  i1 α i  β i − 1   1 2.9 which implies that F α 1 , ,α n ∈Nρ,whereρ  1   n i1 α i β i − 1. Corollary 2.6. Let f i ∈Nβ for each i  1, 2, 3, ,n with β>1.ThenF α 1 , ,α n z ∈Nη with η  1   n i1 α i β − 1 and α i > 0, i  1, 2, 3, ,n. Proof. In Thorem 2.5, we consider β 1  β 2  ···  β n  β. Corollary 2.7. Let f ∈Nβ with β>1. Then the integral operator F α z  z 0  f  t  α dt 2.10 is in the class Nαβ − 11 and α>0. 4 Journal of Inequalities and Applications Proof. We have zF  α z F  α z  α zf  z f  z . 2.11 From 2.11 we have Re  zF  α z F  α z  1   α Re  zf  z f  z  1   1 − α<αβ 1 − α  αβ − 11. 2.12 So the relation 2.12 implies that the operator F α is in Nαβ − 11. Example 2.8. Let fz1/2β − 1{1 − 1 − z 2β−1 }∈Nβ. After some calculi, we obtain that F α z  z 0  f  t  α dt  1 2α1 − β − 1 1 − z 2αβ−11 ∈N  αβ − 11  . 2.13 Acknowledgment The paper is supported by Grant no. 2-CEx 06-11-10/25.07.2006. References 1 B. A. Uralegaddi, M. D. Ganigi, and S. M. Sarangi, “Univalent functions with positive coefficients,” Tamkang Journal of Mathematics, vol. 25, no. 3, pp. 225–230, 1994. 2 S. Owa and H. M. Srivastava, “Some generalized convolution properties associated with certain sub- classes of analytic functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 3, Article ID 42, 13 pages, 2002. 3 D. Breaz and N. Breaz, “Two integral operators,” Studia Universitatis Babes¸-Bolyai, Mathematica, vol. 47, no. 3, pp. 13–19, 2002. 4 D. Breaz, S. Owa, and N. Breaz, “A new integral univalent operator,” in press. . distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let U  {z ∈ C, |z| < 1} be the open unit disk and let A denote the class of the functions. 13 September 2007; Revised 21 October 2007; Accepted 2 January 2008 Recommended by Vijay Gupta We consider the classes M β i  and N β i  of the analytic functions and two general integral opera- tors Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 719354, 4 pages doi:10.1155/2008/719354 Research Article Certain Integral Operators on the Classes M β i 