Research Article Some Properties of Certain Integral Operators on New Subclasses of Analytic Functions with Complex Order

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1 Alied Mathematics Volume 2012, Article ID , 9 ages doi: /2012/ esearch Article Some Proerties of Certain Integral Oerators on New Subclasses of Analytic Functions with Comlex Order Aabed Mohammed and Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia Corresondence should be addressed to Maslina Darus, maslina@ukm.my eceived 15 July 2012; Acceted 15 Setember 2012 Academic Editor: oberto Natalini Coyright q 2012 A. Mohammed and M. Darus. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. We define new subclasses of -valent meromorhic functions with comlex order. We rove some roerties for certain integral oerators on these subclasses. 1. Introduction Let U z C : z < 1} be the oen unit disc in the comlex lane C, U U \0}, the unctured oen unit disk. Let Σ denote the class of meromorhic functions of the form f 1 z a n z n, n 0 N ), 1.1 which are analytic and -valent in U. For 1, we obtain the class of meromorhic functions Σ. We say that a function f Σ is the meromorhic -valent starlike of order α 0 α< and belongs to the class f Σ α, if it satisfies the inequality zf ) >α. f 1.2

2 2 Alied Mathematics A function f Σ is the meromorhic -valent convex function of order α 0 α<, iff satisfies the following inequality: 1 zf ) f >α, 1.3 and we denote this class by ΣK α. Many imortant roerties and characteristics of various interesting subclasses of the class Σ of meromorhically -valent functions were investigated extensively by among others Uralegaddi and Somanatha 1, 2, Liu and Srivastava 3, 4, Mogra 5, 6, Srivastava et a1. 7, Aouf et al. 8, 9, Joshi and Srivastava 10, Owa et a1. 11, and Kulkarni et al. 12. Now, for f Σ, we define the following new subclasses. Definition 1.1. Let a function f Σ be analytic in U. Then f is in the class Σ M β, b if it satisfies the inequality 1 zf )} b f <β, 1.4 N, b C \ 0}, β >. 1.5 Definition 1.2. Let a function f Σ be analytic in U. Then f is in the class Σ N β, b if it satisfies the inequality 1 zf )} b f 1 <β, 1.6 N, b C \ 0}, β >. 1.7 Definition 1.3. Let a function f Σ be analytic in U. Then f is in the class ΣF β, b if it satisfies the inequality 1 z zf 2 ) f ) )} b zf 1 ) <β, f 1.8 N, b C \ 0}, β >. 1.9 For 1 in Definitions 1.1, 1.2, and 1.3, we obtain the following new subclasses of meromorhic functions Σ.

3 Alied Mathematics 3 Definition 1.4. Let a function f Σ be analytic in U. Then f is in the class Σ M β, b if it satisfies the inequality 1 1 zf )} b f 1 <β, 1.10 b C \ 0}, β > Definition 1.5. Let a function f Σ be analytic in U. Then f is in the class Σ N β, b if it satisfies the inequality 1 1 zf )} b f 2 <β, 1.12 b C \ 0}, β > For b 1 in Definition 1.5, weobtainσ N β, the class of meromorhic function, introduced and studied by Wang et al see Definition 1.6. Let a function f Σ be analytic in U. Then f is in the class ΣF β, b if it satisfies the inequality 1 1 z zf 3f ) )} b zf 1 <β, 2f 1.14 b C \ 0}, β > Most recently, Mohammed and Darus 19 introduced the following two general integral oerators of -valent meromorhic functions Σ : 1 z 1 z 0 u f 1 u ) γ1 u f n u ) γ n du; z 1 z 0 ) γ1 ) γn u 1 u f 1 1 u f n u du, 1.17

4 4 Alied Mathematics n, N, 1, 2, 3,...,n}, γ > For 1 in 1.16 and 1.17, resectively, we obtain the general integral oerators F 1,γ H and J 1,γ H γ, introduced by the authors in 16, Main esults In this section, considering the above new subclasses we obtain for the integral oerators F 1,γ and J 1,γ some sufficient conditions for a family of functions f i to be in the the above new subclasses. Theorem 2.1. Let f Σ.Iff Σ M β,b,then ΣF μ, b ), 2.1 n, N, 1,..., n}, γ > 0, b C \ 0}, β >, μ γ β ). 2.2 Proof. Adifferentiation of which is defined in 1.16,weget z 1 F,γ 1 ) z z f 1 ) γ1 z f n ) γ n, z 1 F,γ 2 1 ) z F,γ 1 ) z z f γ ) z 1 f [z z f 1 ) γ1 z f n ) γ n ], f Then from 2.3,weobtain z 1 F,γ 2 1 ) z F,γ 1 ) z 1 z 1 F,γ 1 ) z f γ f ). z 2.4

5 Alied Mathematics 5 By multilying 2.4 with z yield, z 1 F,γ 2 1 ) z F,γ 1 ) z 1 z F,γ 1 ) z 1 zf γ ) f. 2.5 That is equivalent to z 1 F,γ 2 ) z F,γ zf z F,γ 1 ) z 1 γ ) f ; z zf,γ 2 ) ) F,γ zf,γ 1 ) zf γ ) f. 2.6 Equivalently, the above can be written as 1 z zf,γ 2 ) ) F,γ b zf,γ 1 ) γ 1 zf )} b f γ. 2.7 Taking the real art of both terms of 2.7, we have 1 z zf,γ 2 ) ) F,γ b zf,γ 1 ) γ 1 zf )} b f γ. 2.8 Sine f Σ M β,b,weget 1 z zf,γ 2 ) ) F,γ b zf,γ 1 ) < γ β γ. 2.9

6 6 Alied Mathematics That is, 1 z zf,γ 2 ) ) F,γ b zf,γ 1 ) < γ β ) Then ΣF μ, b ), μ γ β ) This comletes the roof. Theorem 2.2. Let f Σ.Iff Σ N β,b,then ΣF μ, b ), 2.12 n, N, 1,..., n}, γ > 0, b C \ 0}, β >, μ γ β ) Proof. Adifferentiation of, which is defined in 1.17, weget z 1 J,γ 1 ) ) γ1 ) γn z z 1 z f 1 1 f n, z 1 J,γ 2 1 ) z J,γ 1 ) z 1 z 1 f γ 1 ) z f )[ ) γ1 ) γn ] z 1 z z 1 f f 1 1 f n Then from 2.14, weobtain z 1 J,γ 2 1 ) z J,γ 1 ) z 1 z J,γ 1 ) z 1 zf γ ) f

7 Alied Mathematics 7 That is equivalent to z zj,γ 2 ) ) J,γ zj,γ 1 ) zf γ ) f Equivalently, the above can be written as 1 z zj,γ 2 ) ) J,γ b zj,γ 1 ) γ 1 b zf f 1 )} γ Taking the real art of both terms of 2.17, we have 1 z zj,γ 2 ) ) J,γ b zj,γ 1 ) γ 1 b zf f 1 )} γ Sine f Σ N β,b,weget 1 z zj,γ 2 ) ) J,γ b zj,γ 1 ) < γ β γ That is, 1 z zj,γ 2 ) ) J,γ b zj,γ 1 ) < γ β ) Then ΣF μ, b ), μ γ β ) This comletes the roof. Putting 1 in Theorem 2.1,wehavethefollowing.

8 8 Alied Mathematics Theorem 2.3. Let f Σ. Iff Σ M β,b,then H n ΣF μ, b ), 2.22 n N, 1,...,n}, γ > 0, b C \ 0}, β > 1, μ 1 γ β 1 ) Putting 1 in Theorem 2.2,wehavethefollowing. Theorem 2.4. Let f Σ. Iff Σ N β,b,then H γ ΣF μ, b ), 2.24 n N, 1,...,n}, γ > 0, b C \ 0}, β > 1, μ 1 γ β 1 ) for other work that we can look at regarding integral oerators see 17, 21, 22. Acknowledgment This work was suorted by UKM-ST-06-FGS and LGS/TD/2011/UKM/ ICT/03/02. eferences 1 B. A. Uralegaddi and C. Somanatha, New criteria for meromorhic starlike univalent functions, Bulletin of the Australian Mathematical Society, vol. 43, no. 1, , B. A. Uralegaddi and C. Somanatha, Certain classes of meromorhic multivalent functions, Tamkang Mathematics, vol. 23, no. 3, , J.-L. Liu and H. M. Srivastava, A linear oerator and associated families of meromorhically multivalent functions, Mathematical Analysis and Alications, vol. 259, no. 2, , J.-L. Liu and H. M. Srivastava, Some convolution conditions for starlikeness and convexity of meromorhically multivalent functions, Alied Mathematics Letters, vol. 16, no. 1, , M. L. Mogra, Meromorhic multivalent functions with ositive coefficients. I, Mathematica Jaonica, vol. 35, no. 1,. 1 11, M. L. Mogra, Meromorhic multivalent functions with ositive coefficients. II, Mathematica Jaonica, vol. 35, no. 6, , H. M. Srivastava, H. M. Hossen, and M. K. Aouf, A unified resentation of some classes of meromorhically multivalent functions, Comuters & Mathematics with Alications, vol. 38, no , , 1999.

9 Alied Mathematics 9 8 M. K. Aouf and H. M. Hossen, New criteria for meromorhic -valent starlike functions, Tsukuba Mathematics, vol. 17, no. 2, , M. K. Aouf and M. H. Srivastava, Anew criterion for meromorhically -valent convex functions of order alha, Mathematical Sciences esearch Hot-Line, vol. 1, no. 8,. 7 12, S. B. Joshi and H. M. Srivastava, A certain family of meromorhically multivalent functions, Comuters & Mathematics with Alications, vol. 38, no. 3-4, , S. Owa, H. E. Darwish, and M. K. Aouf, Meromorhic multivalent functions with ositive and fixed second coefficients, Mathematica Jaonica, vol. 46, no. 2, , S.. Kulkarni, U. H. Naik, and H. M. Srivastava, A certain class of meromorhically -valent quasiconvex functions, Panamerican Mathematical Journal, vol. 8, no. 1, , Z.-G. Wang, Z.-H. Liu, and.-g. Xiang, Some criteria for meromorhic multivalent starlike functions, Alied Mathematics and Comutation, vol. 218, no. 3, , Z.-G. Wang, Y. Sun, and Z.-H. Zhang, Certain classes of meromorhic multivalent functions, Comuters & Mathematics with Alications, vol. 58, no. 7, , Z.-G. Wang, Z.-H. Liu, and A. Cǎtaş, On neighborhoods and artial sums of certain meromorhic multivalent functions, Alied Mathematics Letters, vol. 24, no. 6, , A. Mohammed and M. Darus, A new integral oerator for meromorhic functions, Acta Universitatis Aulensis, no. 24, , A. Mohammed and M. Darus, Integral oerators on new families of meromorhic functions of comlex order, Inequalities and Alications, vol. 2011, article 121, 12 ages, B. A. Frasin, On an integral oerator of meromorhic functions, Matematichki Vesnik, vol. 64, no. 2, , A. Mohammed and M. Darus, The order of starlikeness of new -valent meromorhic functions, International Mathematical Analysis, vol. 6, no. 27, , A. Mohammed and M. Darus, Starlikeness roerties of a new integral oerator for meromorhic functions, Alied Mathematics, vol. 2011, Article ID , 8 ages, A. Mohammed and M. Darus, New roerties for certain integral oerators, International Mathematical Analysis, vol. 4, no , , N. Breaz, D. Breaz, and M. Darus, Convexity roerties for some general integral oerators on uniformly analytic functions classes, Comuters & Mathematics with Alications, vol. 60, no. 12, , 2010.

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