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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 452057, 10 pages doi:10.1155/2008/452057 Research Article A New Subclass of Analytic Functions Involving Al-Oboudi Differential Operator Sevtap S ¨ umer Eker and H. ¨ Ozlem G ¨ uney Department of Mathematics, Faculty of Science and Letters, Dicle University, 21280 Diyarbakir, Turkey Correspondence should be addressed to Sevtap S ¨ umer Eker, sevtaps@dicle.edu.tr Received 25 September 2007; Accepted 4 February 2008 Recommended by Jozsef Szabados The main object of this paper is to introduce and investigate a new subclass of normalized analytic functions in the open unit disc U which is defined by Al-Oboudi differential operator. Coefficient inequalities, extreme points, and integral means inequalities for fractional derivative for this class are given. Copyright q 2008 S. S ¨ umer Eker and H. ¨ Ozlem G ¨ uney. 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 and definitions Let A denote the class of functions f normalized by fzz  ∞  j2 a j z j 1.1 which are analytic in the open unit disc U  {z : |z| < 1}. For f ∈A, Al-Oboudi 1 introduced the following operator: D 0 fzfz, 1.2 D 1 fz1 − δfzδzf  zD δ fz,δ≥ 0 1.3 D n fzD δ  D n−1 fz  , n ∈ N  1, 2, 3, . 1.4 If f is given by 1.1,thenfrom1.3 and 1.4 we see that D n fzz  ∞  j2 1 j − 1δ n a j z j ,  n ∈ N 0  N ∪{0}  . 1.5 When δ  1, we get Sˇalˇagean differential operator 2. 2 Journal of Inequalities and Applications Definition 1.1. Let S m,n,δ α denote the subclass of A consisting of functions f which satisfy the inequality Re  D m fz D n fz  >α 1.6 for some 0 ≤ α<1, m ∈ N, n ∈ N 0 ,andallz ∈ U. The object of the present paper is to investigate the coefficient bounds, extreme points, and integral mean inequalities for fractional derivatives of functions belonging to the class S m,n,δ α. 2. Coefficient inequalities Our first theorem gives a sufficient condition for f ∈Ato belong to the class S m,n,δ α. Theorem 2.1. Let fz ∈Asatisfy ∞  j2 Ψm, n, j, δ, α|a j |≤21 − α, 2.1 where Ψm, n, j, δ, α   1j−1δ m −1α1j −1δ n   1j−1δ m 1−α1j −1δ n 2.2 for some α 0 ≤ α<1, m ∈ N, n ∈ N 0 , δ δ ≥ 0.Thenfz ∈S m,n,δ α. Proof. Suppose that 2.1 is true for α0 ≤ α<1, m ∈ N, n ∈ N 0 ,andδδ ≥ 0.Forfz ∈A, define the function Fz by Fz D m fz D n fz − α. 2.3 It suffices to show that     Fz − 1 Fz1     < 1 z ∈ U. 2.4 We note that     Fz − 1 Fz1          D m fz/D n fz − α − 1 D m fz/D n fz − α  1          D m fz − 1  αD n fz D m fz1 − αD n fz           α −  ∞ j2  1 j − 1δ m − 1  α1 j − 1δ n  a j z j−1 2 − α  ∞ j2  1 j − 1δ m 1 − α1 j − 1δ n  a j z j−1      ≤ α   ∞ j2   1 j − 1δ m − 1  α1 j − 1δ n   |a j ||z| j−1 2 − α −  ∞ j2  1 j − 1δ m 1 − α1 j − 1δ n  |a j ||z| j−1 < α   ∞ j2   1 j − 1δ m − 1  α1 j − 1δ n   |a j | 2 − α −  ∞ j2  1 j − 1δ m 1 − α1 j − 1δ n  |a j | . 2.5 S. S ¨ umer Eker and H. ¨ Ozlem G ¨ uney 3 The last expression is bounded above by 1 if α  ∞  j2   1 j − 1δ m − 1  α1 j − 1δ n   |a j | ≤ 2 − α − ∞  j2  1 j − 1δ m 1 − α1 j − 1δ n  |a j | 2.6 which is equivalent to condition 2.1. This completes the proof of Theorem 2.1. Example 2.2. The function fz given by fzz  ∞  j2 22  γ1 − α j j  γj  1  γΨm, n, j, δ, α z j 2.7 belongs to the class S m,n,δ α for γ>−2, 0 ≤ α<1,  j ∈ C,and| j |  1. We now derive the coefficient inequalities for fz belonging to the class S m,n,δ α. Theorem 2.3. If fz ∈S m,n,δ α,thenfork ≥ 2,   a k   ≤ β |v k |  1  β k−1  j2 1 j − 1δ n   v j    β 2 k−1  j 2 >j 1 k−2  j 1 2  1   j 1 − 1  δ  1 j 2 − 1δ  n   v j 1 v j 2    β 3 k−1  j 3 >j 2 k−2  j 2 >j 1 k−3  j 1 2  1   j 1 − 1  δ  1   j 2 − 1  δ  1   j 3 − 1  δ  n   v j 1 v j 2 v j 3    ···  β k−2 k−1  j2 1 j − 1δ n   v j |  , 2.8 where β  21 − α and v k 1 k − 1δ m − 1 k − 1δ n . Proof. Define the function pz by pz 1 1 − α  D m fz D n fz − α   1  ∞  j1 c j z j . 2.9 Since pz is the Carath ´ eodory function, we have that |c j |≤2 j  1, 2, 3, . 2.10 The definition of pz implies that 1 1 − α  D m fz − αD n fz   D n fz  1  ∞  j1 c j z j  . 2.11 4 Journal of Inequalities and Applications Since D n fzz  ∞  j2  1 j − 1δ  n a j z j  n ∈ N 0  , 2.12 we have D m fz − αD n fz 1 − α  z  1  δ m − α1  δ n 1 − α a 2 z 2  1  2δ m − α1  2δ n 1 − α a 3 z 3  ···  1 k − 1δ m − α1 k − 1δ n 1 − α a k z k  ··· , D n fz  1  ∞  j1 c j z j    z  ∞  j2  1 j − 1δ  n a j z j   1  c 1 z  ··· c k z k  ···  . 2.13 Therefore, 2.11 shows that z 1δ m −α1δ n 1−α a 2 z 2  12δ m −α12δ n 1−α a 3 z 3 ··· 1k−1δ m −α1k−1δ n 1−α a k z k ···   z  ∞  j2  1 j − 1δ  n a j z j   1  c 1 z  ··· c k z k  ···  . 2.14 If we consider the coefficients of z k of the both sides in the above equality, then we find that   1 k − 1δ  m − α  1 k − 1δ  n 1 − α −  1 k − 1δ  n  a k  k−1  j1  1 k − j − 1δ  n a k−j c j . 2.15 Therefore,   a k    1 − α   1 k − 1δ m − 1 k − 1δ n        k−1  j1 1 k − j − 1δ n a k−j c j      ≤ 1 − α   1 k − 1δ m − 1 k − 1δ n    k−1  j1 1 k − j − 1δ n   a k−j     c j    ≤ 21 − α   1 k − 1δ m − 1 k − 1δ n    k−1  j1 1 k − j − 1δ n   a k−j    , 2.16 S. S ¨ umer Eker and H. ¨ Ozlem G ¨ uney 5 since |c j |≤2 j  1, 2, 3 . Thus, for β  21 − α and v k 1 k − 1δ m − 1 k − 1 δ n ,we obtain |a k |≤β 1 |v k |  1 1  δ n β   v 2   1  2δ n β   v 3   1  3δ n β   v 4    ···1 k − 2δ n β   v k−1   1  δ n 1  2δ n β 2   v 2 v 3   1  δ n 1  3δ n β 2   v 2 v 4   1  δ n 1  4δ n β 2   v 2 v 5    ···1  δ n 1 k − 2δ n β 2   v 2 v k−1   1  2δ n 1  3δ n β 2   v 3 v 4   1  2δ n 1  4δ n β 2   v 3 v 5    ··· 1  2δ n 1 k − 2δ n β 2   v 3 v k−1    ··· 1  δ n 1  2δ n 1  3δ n β 3   v 2 v 3 v 4   1  δ n 1  3δ n 1  4δ n β 3   v 2 v 4 v 5    ··· 1  δ n 1 k − 3δ n 1 k − 2δ n β 3   v 2 v k−2 v k−1    β k−2 k−1  j2 1 j − 1δ n   v j     β |v k |  1 β k−1  j2 1 j − 1δ n   v j    β 2 k−1  j 2 >j 1 k−2  j 1 2 1 j 1 − 1δ1 j 2 − 1δ n   v j 1 v j 2    β 3 k−1  j 3 >j 2 k−2  j 2 >j 1 k−3  j 1 2  1  j 1 −1  δ, 1  j 2 −1  δ, 1  j 3 −1  δ   n   v j 1 v j 2 v j 3   ···β k−2 k−1  j2  1j−1δ  n   v j    . 2.17 This completes the proof of Theorem 2.3. If we take δ  1 in Theorems 2.1 and 2.3, we can get the results due to S ¨ umer Eker and Owa 3. 3. Extreme points In view of Theorem 2.1, we now introduce the subclass  S m,n,δ α ⊂S m,n,δ α, which consists of function fzz  ∞  j2 a j z j a j ≥ 03.1 whose Taylor-Maclaurin coefficients satisfy inequality 2.1. Now, let us determine extreme points of the class  S m,n,δ α. 6 Journal of Inequalities and Applications Theorem 3.1. Let f 1 zz and f j zz  21 − α Ψm, n, j, δ, α z j j  2, 3, , 3.2 where Ψm, n, j, δ, α is given by 2.2. Then f ∈  S m,n α if and only if it can be expressed in the form fz ∞  j1 η j f j z, 3.3 where η j > 0 and  ∞ j1 η j  1. Proof. Suppose that fz ∞  j1 η j f j zz  ∞  j2 η j 21 − α Ψm, n, j, δ, α z j . 3.4 Then ∞  j2 Ψm, n, j, δ, α 21 − α Ψm, n, j, δ, α η j  21 − α ∞  j2 η j  21 − α1 − η 1  < 21 − α, 3.5 which shows that f satisfies condition 2.1 and therefore f ∈  S m,n,δ α. Conversely, suppose that f ∈  S m,n,δ α. Since a j ≤ 21 − α Ψm, n, j, δ, α j  2, 3, , 3.6 we may set η j  Ψm, n, j, δ, α 21 − α a j , η 1  1 − ∞  j2 η j . 3.7 Then we obtain fz ∞  j1 η j f j z, 3.8 which completes the proof of Theorem 3.1. Corollary 3.2. The extreme points of  S m,n,δ α are the functions f 1 zz and f j zz  21 − α Ψm, n, j, δ, α z j j  2, 3, , 3.9 where Ψm, n, j, δ, α is given by 2.2. S. S ¨ umer Eker and H. ¨ Ozlem G ¨ uney 7 4. Integral means inequalities for fractional derivative We will make use of the following definitions of fractional derivatives by Owa 4,andSrivas- tava and Owa 5. Definition 4.1. The fractional derivative of order λ is defined, for a function f,by D λ z fz 1 Γ1 − λ d dz  z 0 fξ z − ξ λ dξ 0 ≤ λ<1, 4.1 where f is an analytic function in a simply connected region of z-plane containing the origin, and the multiplicity of z − ξ −λ is removed by requiring logz − ξ to be real when z − ξ>0. Definition 4.2. Under the hypotheses of Definition 4.1, the fractional derivative of order p  λ is defined, for a function f,by D pλ z fz d p dz p D λ z fz0 ≤ λ<1; p ∈ N 0 . 4.2 It readily follows from 4.1 that D λ z z k  Γk  1 Γk − λ  1 z k−λ 0 ≤ λ<1,k ∈ N. 4.3 Further, we need the concept of subordination between analytic functions 6 and a subordi- nation theorem of Littlewood in our investigation. Definition 4.3. For two functions f and g, analytic in U, say that the function fz is subordinate to gz in U,andwrite fz ≺ gzz ∈ U4.4 if there exists a Schwarz function wz, analytic in U with w00and|wz| < 1 such that fzgwz z ∈ U. 4.5 In particular, if the function g is univalent in U, the above subordination is equivalent to f0g0,f U ⊂ gU. 4.6 In 1925, Littlewood 7 proved the following subordination theorem. Lemma 4.4. If fz and gz are analytic in U with fz≺ g(z), then for μ>0 and z re iθ 0<r<1,  2π 0 |fz| μ dθ   2π 0 |gz| μ dθ. 4.7 8 Journal of Inequalities and Applications Theorem 4.5. Let fz ∈  S m,n,δ α and suppose that ∞  j2 j − p p1 a j ≤ 21 − αΓk  1Γ3 − λ − p Ψm, n, k, δ, αΓk  1 − λ − pΓ2 − p 4.8 for some j ≥ p, 0 ≤ λ<1,andj − p p1 denote the Pochhammer symbol defined by j − p p1  j − pj − p  1 ···j. Also let the function f k zz  21 − α Ψm, n, k, δ, α z k k ≥ 2. 4.9 If there exists an analytic function wz given by wz k−1  Ψm, n, k, δ, αΓk  1 − λ − p 21 − αΓk  1 ∞  j2 j − p p1 Γj − p Γj  1 − λ − p a j z j−1 , k ≥ p, 4.10 then for z  re iθ and 0 <r<1,  2π 0   D pλ z fz   μ dθ ≤  2π 0   D pλ z f k z   μ dθ 0 ≤ λ<1,μ >0. 4.11 Proof. By virtue of the fractional derivative formula 4.3 and Definition 4.2, we find from 3.1 that D pλ z fz z 1−λ−p Γ2 − λ − p  1  ∞  j2 Γ2 − λ − pΓj  1 Γj  1 − λ − p a j z j−1   z 1−λ−p Γ2 − λ − p  1  ∞  j2 Γ2 − λ − pj − p p1 Φja j z j−1  , 4.12 where Φj Γj − p Γj  1 − λ − p 0 ≤ λ<1; j ≥ p. 4.13 Since Φj is a decreasing function of j,wehave 0 < Φj ≤ Φ2 Γ2 − p Γ3 − λ − p . 4.14 Similarly, from 4.3, 4.9,andDefinition 4.2,weobtain D pλ z f k z z 1−λ−p Γ2 − λ − p  1  21 − αΓ2 − λ − pΓk  1 Ψm, n, k, δ, αΓk  1 − λ − p z k−1  . 4.15 S. S ¨ umer Eker and H. ¨ Ozlem G ¨ uney 9 For z  re iθ ,0<r<1, we must show that  2π 0      1  ∞  j2 Γ2 − λ − pj − p p1 Φja j z j−1      μ dθ ≤  2π 0     1  21 − αΓ2 − λ − pΓk  1 Ψm, n, k, δ, αΓk  1 − λ − p z k−1     μ dθ μ>0. 4.16 Thus by applying Littlewood’s subordination theorem, it would be suffice to show that 1  ∞  j2 Γ2 − λ − pj − p p1 Φja j z j−1 ≺ 1  21 − αΓ2 − λ − pΓk  1 Ψm, n, k, δ, αΓk  1 − λ − p z k−1 . 4.17 By setting 1  ∞  j2 Γ2 − λ − pj − p p1 Φja j z j−1  1  21 − αΓ2 − λ − pΓk  1 Ψm, n, k, δ, αΓk  1 − λ − p wz k−1 , 4.18 we find that wz k−1  Ψm, n, k, δ, αΓk  1 − λ − p 21 − αΓk  1 ∞  j2 j − p p1 Φja j z j−1 4.19 which readily yields w00. Further, we prove that the analytic function wz satisfies |wz| < 1, z ∈ U for 4.10. We know that |wz| k−1 ≤      Ψm, n, k, δ, αΓk  1 − λ − p 21 − αΓk  1 ∞  j2 j − p p1 Φja j z j−1      ≤ Ψm, n, k, δ, αΓk  1 − λ − p 21 − αΓk  1 ∞  j2 j − p p1 Φja j |z| j−1 ≤|z| Ψm, n, k, δ, αΓk  1 − λ − p 21 − αΓk  1 Φ2 ∞  j2 j − p p1 a j  |z| Ψm, n, k, δ, αΓk  1 − λ − p 21 − αΓk  1 Γ2 − p Γ3 − λ − p ∞  j2 j − p p1 a j ≤|z| < 1 4.20 by means of the hypothesis of Theorem 4.5. As special case p  0, Theorem 4.5 readily yields. Corollary 4.6. Let fz ∈  S m,n,δ α and suppose that ∞  j2 ja j ≤ 21 − αΓk  1Γ3 − λ Ψm, n, k, δ, αΓk  1 − λ 4.21 10 Journal of Inequalities and Applications for some 0 ≤ λ<1. Also let the function f k zz  21 − α Ψm, n, k, δ, α z k k ≥ 2. 4.22 If there exists an analytic function wz given by wz k−1  Ψm, n, k, δ, αΓk  1 − λ 21 − αΓk  1 ∞  j2 Γj  1 Γj  1 − λ a j z j−1 , 4.23 then for z  re iθ and 0 <r<1,  2π 0   D λ z fz   μ dθ ≤  2π 0   D λ z f k z   μ dθ 0 ≤ λ<1,μ >0. 4.24 Acknowledgment The authors are thankful to the referees for their comments and suggestions. References 1 F. M. Al-Oboudi, “On univalent functions defined by a generalized S ˘ al ˘ agean operator,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 27, pp. 1429–1436, 2004. 2 G. S. S ˘ al ˘ agean, “Subclasses of univalent functions,” in Complex Analysis—5th Romanian-Finnish seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin, Germany, 1983. 3 S. S ¨ umer Eker and S. Owa, “New applications of classes of analytic functions involving the S ˘ al ˘ agean operator,” in Proceedings of the International Symposium on Complex Function Theory and Applications, pp. 21–34, Transilvania University of Printing House, Bras¸ov, Romania, September 2006. 4 S. Owa, “On the distortion theorems. I,” Kyungpook Mathematical Journal, vol. 18, no. 1, pp. 53–59, 1978. 5 H. M. Srivastava and S. Owa, Eds., Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK, 1989. 6 P. L. Du r e n, Univalent Functions, Springer, New York, NY, USA, 1983. 7 J. E. Littlewood, “On inequalities in the theory of functions,” Proceedings of the London Mathematical Society, vol. 23, no. 1, pp. 481–519, 1925. . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 452057, 10 pages doi:10.1155/2008/452057 Research Article A New Subclass of Analytic Functions Involving Al-Oboudi. comments and suggestions. References 1 F. M. Al-Oboudi, “On univalent functions defined by a generalized S ˘ al ˘ agean operator,” International Journal of Mathematics and Mathematical Sciences,. Berlin, Germany, 1983. 3 S. S ¨ umer Eker and S. Owa, New applications of classes of analytic functions involving the S ˘ al ˘ agean operator,” in Proceedings of the International Symposium