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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 158408, 13 pages doi:10.1155/2009/158408 Research Article A New General Integral Operator Defined by Al-Oboudi Differential Operator Serap Bulut ˙ Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Izmit-Kocaeli, Turkey Correspondence should be addressed to Serap Bulut, serap.bulut@kocaeli.edu.tr Received December 2008; Accepted 22 January 2009 Recommended by Narendra Kumar Govil We define a new general integral operator using Al-Oboudi differential operator Also we introduce new subclasses of analytic functions Our results generalize the results of Breaz, Guney, ă and S˘ l˘ gean aa Copyright q 2009 Serap Bulut 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 Introduction Let A denote the class of functions of the form f z z ∞ an zn 1.1 n which are analytic in the open unit disk U {z ∈ C : |z| < 1}, and S : {f ∈ A : f is univalent in U} For f ∈ A, Al-Oboudi introduced the following operator: D0 f z D1 f z Dk f z 1−λ f z f z, λzf z Dλ Dk−1 f z , 1.2 Dλ f z , λ ≥ 0, 1.3 k ∈ N : {1, 2, 3, } 1.4 Journal of Inequalities and Applications If f is given by 1.1 , then from 1.3 and 1.4 we see that Dk f z ∞ z n − λ k an zn , k ∈ N0 : N ∪ {0} , 1.5 n with Dk f 0 Remark 1.1 When λ 1, we get S˘ l˘ gean’s differential operator aa Now we introduce new classes Sk δ, b, λ and Kk δ, b, λ as follows A function f ∈ A is in the classes Sk δ, b, λ , where δ ∈ 0, , b ∈ C − {0}, λ ≥ 0, k ∈ N0 , if and only if b Re Dk f z −1 Dk f z >δ 1.6 or equivalently Re λ b z Dk f z Dk f z −1 >δ 1.7 for all z ∈ U A function f ∈ A is in the classs Kk δ, b, λ , where δ ∈ 0, , b ∈ C − {0}, λ ≥ 0, k ∈ N0 , if and only if Re λ z Dk f z b Dk f z >δ 1.8 for all z ∈ U We note that f ∈ Kk δ, b, λ if and only if zf ∈ Sk δ, b, λ Remark 1.2 i For k and λ 1, we have the classes ∗ S0 δ, b, ≡ Sδ b , introduced by Frasin ii For b and λ K0 δ, b, ≡ Cδ b 1.9 1, we have the class Sk δ, 1, ≡ Sk δ 1.10 of k-starlike functions of order δ defined by S˘ l˘ gean aa iii In particular, the classes S0 δ, 1, ≡ S∗ δ , K0 δ, 1, ≡ K δ 1.11 Journal of Inequalities and Applications are the classes of starlike functions of order δ and convex functions of order δ in U, respectively iv Furthermore, the classes S0 0, 1, ≡ S∗ , K0 0, 1, ≡ K 1.12 are familiar classes of starlike and convex functions in U, respectively v For λ 1, we get Kk δ, b, ≡ Sk δ, b, 1.13 Let us introduce the new subclasses USk α, δ, b, λ , UKk α, δ, b, λ and SHk α, b, λ , KHk α, b, λ as follows A function f ∈ A is in the class USk α, δ, b, λ if and only if f satisfies Re Dk f z −1 Dk f z b >α b Dk f z −1 Dk f z δ z∈U 1.14 −1 δ, 1.15 or equivalently Re λ b z Dk f z Dk f z −1 >α λ b z Dk f z Dk f z where α ≥ 0, δ ∈ −1, , α δ ≥ 0, b ∈ C − {0}, λ ≥ 0, k ∈ N0 A function f ∈ A is in the class UKk α, δ, b, λ if and only if f satisfies Re λ z Dk f z b Dk f z >α λ z Dk f z b Dk f z δ, 1.16 where α ≥ 0, δ ∈ −1, , α δ ≥ 0, b ∈ C − {0}, λ ≥ 0, k ∈ N0 We note that f ∈ UKk α, δ, b, λ if and only if zf ∈ USk α, δ, b, λ Remark 1.3 i For α 0, we have USk 0, δ, b, λ ≡ Sk δ, b, λ , ii For b and λ UKk 0, δ, b, λ ≡ Kk δ, b, λ 1.17 1, we have the class USk α, δ, 1, ≡ USk α, δ 1.18 of k-uniform starlike functions of order δ and type α, iii For λ 1, we have UKk α, δ, b, ≡ USk α, δ, b, 1.19 Journal of Inequalities and Applications iv For b and λ 1, we have UKk α, δ, 1, ≡ USk α, δ 1.20 Geometric Interpretation f ∈ USk α, δ, b, λ and f ∈ UKk α, δ, b, λ if and only if λ/b z Dk f z /Dk f z − and λ/b z Dk f z / Dk f z , respectively, take all the values in the conic domain Rα,δ which is included in the right-half plane such that Rα,δ u iv : u > α u−1 v2 δ 1.21 From elementary computations we see that ∂Rα,δ represents the conic sections symmetric about the real axis Thus Rα,δ is an elliptic domain for α > 1, a parabolic domain for α 1, a hyperbolic domain for < α < and a right-half plane u > δ for α A function f ∈ A is in the class SHk α, b, λ if and only if f satisfies 1 b √ Dk f z − − 2α − Dk f z √ z∈U , 2α − < Re √ Dk f z −1 Dk f z b 1.22 where α > 0, b ∈ C − {0}, λ ≥ 0, k ∈ N0 A function f ∈ A is in the class KHk α, b, λ if and only if f satisfies √ λ z Dk f z − 2α − b Dk f z √ 2α − z∈U , < Re √ λ z Dk f z b Dk f z 1.23 where α > 0, b ∈ C − {0}, λ ≥ 0, k ∈ N0 We note that f ∈ KHk α, b, λ if and only if zf ∈ SHk α, b, λ Remark 1.4 i For b and λ 1, we have the classes SHk α, 1, ≡ SHk α , KHk α, 1, ≡ SHk defined in ii For λ α, 1, ≡ SHk α 1.24 1, we have KHk α, b, ≡ SHk α, b, 1.25 Journal of Inequalities and Applications D Breaz and N Breaz introduced and studied the integral operator z Fn z μ1 f1 t t ··· fn t t μn dt, 1.26 where fi ∈ A and μi > for all i ∈ {1, , n} By using the Al-Oboudi differential operator, we introduce the following integral operator So we generalize the integral operator Fn l1 , , ln ∈ Nn , and μi > 0, ≤ i ≤ n One defines the integral Definition 1.5 Let k ∈ N0 , l operator Ik,n,l,μ : An → A, Ik,n,l,μ f1 , , fn Dk F z z μ1 Dl1 f1 t t ··· F, μn Dln fn t t 1.27 dt, where f1 , , fn ∈ A and D is the Al-Oboudi differential operator Remark 1.6 In Definition 1.5, if we set i λ 1, then we have 7, Definition ··· ln 0, then we have the integral operator defined ii λ 1, k and l1 by 1.26 iii k 0, l1 · · · ln l ∈ N0 , then we have 8, Definition 1.1 Main Results The following lemma will be required in our investigation Lemma 2.1 For the integral operator Ik,n,l,μ f1 , , fn λz Dk F z Dk F z n i μi F, defined by 1.27 , one has Dli fi z − Dli fi z n μi 2.1 i Proof By 1.27 , we get Dk F z Dl1 f1 z z μ1 ··· Dln fn z z μn 2.2 Also, using 1.3 and 1.4 , we obtain Dk F z Dk F z − − λ Dk F z λz 2.3 Journal of Inequalities and Applications On the other hand, from 2.2 and 2.3 , we find Dk F z n μi i μi Dli fi z z z Dli fi z − Dli fi z zDli fi z j j /i Dk F z − − λ Dk F z λ2 z2 Dk F z n Dlj fj z z − λ Dk F z μj , 2.4 2.5 Thus by 2.2 and 2.4 , we can write Dk F z n Dk F z z Dli fi z i μi n − Dli fi z zDli fi z 2.6 Dli fi z − Dli fi z λzDli fi z μi i Finally, we obtain λz Dk F z n Dk F z μi i Dli fi z −1 , Dli fi z 2.7 which is the desired result Theorem 2.2 Let αi ≥ 0, δi ∈ −1, , αi that δi ≥ ≤ i ≤ n , and b ∈ C − {0}, λ ≥ Also suppose n μi i 1 − δi ≤ 1 αi 2.8 If fi ∈ USli αi , δi , b, λ ≤ i ≤ n , then the integral operator Ik,n,l,μ class Kk γ, b, λ , where γ 1− n i μi − δi αi F, defined by 1.27 , is in the 2.9 Proof Since fi ∈ USli αi , δi , b, λ ≤ i ≤ n , by 1.14 we have Re 1 b Dli fi z −1 Dli fi z > αi δi αi 2.10 Journal of Inequalities and Applications for all z ∈ U By 2.1 , we get 1 λz Dk F z b Dk F z n μi i n Dli fi z −1 Dli fi z b b μi 1 i Dli fi z −1 Dli fi z − n 2.11 μi i So, 2.10 and 2.11 give us Re 1 λz Dk F z b Dk F z 1− n n μi i >1− i n μi Re n μi i i αi δi μi αi 1 for all z ∈ U Hence, we obtain F ∈ Kk γ, b, λ , where γ Corollary 2.3 Let αi ≥ 0, δi ∈ −1, , αi b Dli fi z −1 Dli fi z − δi 1− μi αi i 1− n i μ i − δi / αi δi ≥ ≤ i ≤ n , and b ∈ C − {0} Also suppose that n μi i 1 − δi ≤ 1 αi If fi ∈ USli αi , δi , b, 1 ≤ i ≤ n , then the integral operator Ik,n,l,μ class Sk γ, b, , where γ is defined as in 2.9 Proof In Theorem 2.2, we consider λ 2.12 n 2.13 F, defined by 1.27 , is in the From Corollary 2.3, we immediately get Corollary 2.4 Corollary 2.4 Let αi ≥ 0, δi ∈ −1, , αi δi ≥ ≤ i ≤ n , and b ∈ C − {0} Also suppose that n i If fi ∈ USli αi , δi , b, class Sk 0, b, μi − δi ≤ 1 αi ≤ i ≤ n , then the integral operator Ik,n,l,μ 2.14 F, defined by 1.27 , is in the Remark 2.5 If we set b in Corollary 2.4, then we have 7, Theorem So Corollary 2.4 is an extension of Theorem Corollary 2.6 Let δi ∈ 0, 1 ≤ i ≤ n and b ∈ C − {0}, λ ≥ Also suppose that n i μi − δi ≤ 2.15 Journal of Inequalities and Applications If fi ∈ Sli δi , b, λ ≤ i ≤ n , then the integral operator Ik,n,l,μ Kk ρ, b, λ , where ρ n 1− F, defined by 1.27 , is in the class μ i − δi 2.16 i α2 Proof In Theorem 2.2, we consider α1 Corollary 2.7 Let δi ∈ 0, ··· αn ≤ i ≤ n and b ∈ C − {0} Also suppose that n μi − δi ≤ 2.17 i If fi ∈ Sli δi , b, 1 ≤ i ≤ n , then the integral operator Ik,n,l,μ Sk ρ, b, , where ρ is defined as in 2.16 Proof In Corollary 2.6, we consider λ F, defined by 1.27 , is in the class Corollary 2.8 readily follows from Corollary 2.7 Corollary 2.8 Let δi ∈ 0, 1 ≤ i ≤ n , and b ∈ C − {0} Also suppose that n μi − δi ≤ 2.18 i If fi ∈ Sli δi , b, Sk 0, b, ≤ i ≤ n , then the integral operator Ik,n,l,μ Remark 2.9 If we set b F, defined by 1.27 , is in the class in Corollary 2.8, then we have 7, Corollary Theorem 2.10 Let αi ≥ 0, δi ∈ −1, , αi δi ≥ ≤ i ≤ n and b ∈ C − {0}, λ ≥ Also suppose that n μi ≤ 2.19 i If fi ∈ USli αi , δi , b, λ ≤ i ≤ n , then the integral operator Ik,n,l,μ class Kk γ, b, λ , where γ is defined as in 2.9 F , defined by 1.27 , is in the Proof The proof is similar to the proof of Theorem 2.2 Corollary 2.11 Let αi ≥ 0, δi ∈ −1, , αi δi ≥ ≤ i ≤ n and b ∈ C − {0} Also suppose that n i μi ≤ 2.20 Journal of Inequalities and Applications If fi ∈ USli αi , δi , b, 1 ≤ i ≤ n , then the integral operator Ik,n,l,μ class Sk γ, b, , where γ is defined as in 2.9 Proof In Theorem 2.10, we consider λ Remark 2.12 If we set b F, defined by 1.27 , is in the 1 in Corollary 2.11, then we have 7, Theorem Corollary 2.13 Let δi ∈ 0, 1 ≤ i ≤ n and b ∈ C − {0}, λ ≥ Also suppose that n μi ≤ 2.21 i If fi ∈ Sli δi , b, λ ≤ i ≤ n , then the integral operator Ik,n,l,μ Kk ρ, b, λ , where ρ is defined as in 2.16 Proof In Theorem 2.10, we consider α1 α2 ··· αn F, defined by 1.27 , is in the class Corollary 2.14 Let δi ∈ 0, 1 ≤ i ≤ n and b ∈ C − {0} Also suppose that n μi ≤ 2.22 i If fi ∈ Sli δi , b, 1 ≤ i ≤ n , then the integral operator Ik,n,l,μ Sk ρ, b, , where ρ is defined as in 2.16 Proof In Corollary 2.13, we consider λ Remark 2.15 If we set b F, defined by 1.27 , is in the class 1 in Corollary 2.14, then we have 7, Corollary Theorem 2.16 Let α ≥ 0, δ ∈ −1, , α δ ≥ and b ∈ C − {0}, λ ≥ Also suppose that n μi ≤ 2.23 i If fi ∈ USli α, δ, b, λ ≤ i ≤ n , then the integral operator Ik,n,l,μ class UKk α, δ, b, λ Proof Since fi ∈ USli α, δ, b, λ Re for all z ∈ U b F, defined by 1.27 , is in the ≤ i ≤ n , by 1.14 we have Dli fi z −1 Dli fi z >α b Dli fi z −1 Dli fi z δ 2.24 10 Journal of Inequalities and Applications On the other hand, from 2.1 , we obtain λ z Dk F z b Dk F z n μi i 1− n b Dli fi z −1 Dli fi z n μi i μi i Dli fi z −1 Dli fi z b 2.25 Considering 1.16 with the above equality, we find λ z Dk F z b Dk F z Re 1− n n μi i ≥1− n n n μi μi Re i n μi i 1−δ μi Re i i >1− −α μi α i 1− n μi b λ z Dk F z b Dk F z −δ b Dli fi z −1 Dli fi z b Dli fi z −1 Dli fi z −α n −α Dli fi z −1 Dli fi z b Dli fi z −1 Dli fi z −δ μi b Dli fi z −1 Dli fi z −δ μi b Dli fi z −1 Dli fi z −δ μi i n i δ −α n i ≥0 i 2.26 for all z ∈ U This completes proof Corollary 2.17 Let α ≥ 0, δ ∈ −1, , α δ ≥ 0, and b ∈ C − {0} Also suppose that n μi ≤ 2.27 i If fi ∈ USli α, δ, b, 1 ≤ i ≤ n , then the integral operator Ik,n,l,μ class USk α, δ, b, Proof In Theorem 2.16, we consider λ Remark 2.18 If we set b F, defined by 1.27 , is in the 1 in Corollary 2.17, then we have 7, Theorem Theorem 2.19 Let α ≥ 0, b ∈ C − {0}, and λ ≥ Also suppose that n μi ≤ 2.28 i If fi ∈ SHli α, b, λ class KHk α, b, λ ≤ i ≤ n , then the integral operator Ik,n,l,μ F, defined by 1.27 , is in the Journal of Inequalities and Applications Proof Since fi ∈ SHli α, b, λ Re √ √ ≤ i ≤ n , by 1.22 we have Dli fi z −1 Dli fi z b 11 √ 2α Dli fi z −1 Dli fi z b 2−1 − − 2α √ 2−1 >0 2.29 for all z ∈ U Considering this inequality and 2.1 , we obtain √ √ Re λz Dk F z b Dk F z √ Re √ n μi b i 2α Dli fi z −1 Dli fi z − 1 n μi bi Dli fi z −1 Dli fi z √ n √ i n − i √ n Dli fi z −1 Dli fi z b μi √ μi Re √ n i √ 1− n μi 2α √ n 2−1 i − √ 1− n n 1− 2−1 n μi μi 2α √ 2−1 i − − 2α − √ √ n √ − √ 1− n μi √ 2−1 n i μi − 2α √ 2−1 n − μi i n i μi n √ b b 2α √ 2−1 μi 2α √ n √ Dli fi z −1 Dli fi z Dli fi z −1 Dli fi z √ μi −2α 2−1 i − 2α √ 2−1 Dli fi z −1 Dli fi z 2 b b Dli fi z −1 Dli fi z i 2−1 μi i μi μi Re n i i i 2α 2−1 2−1 μi Re i ≥ √ i √ − 2α √ 2−1 2−1 2α − 2α √ − 2α 2−1 √ Dli fi z −1 −2α 2−1 li f z D i b μi √ Dli fi z −1 Dli fi z b i − − 2α λz Dk F z b Dk F z √ 2α Dli fi z −1 Dli fi z b μi Re √ 2−1 − − 2α √ 2−1 2−1 12 Journal of Inequalities and Applications √ 2α √ − − − 2α √ 1− 2−1 n μi i n √ √ μi Re i > √ 2α √ Dli fi z −1 Dli fi z b − − − 2α √ 2α 1− 2−1 √ n 2−1 − 1 b √ Dli fi z −1 −2α 2−1 Dli fi z μi i > n 1− μi √ √ 4α , 2−1 1 ≥0 i 2.30 for all z ∈ U Hence by 1.23 , we have F ∈ KHk α, b, λ Corollary 2.20 Let α ≥ and b ∈ C − {0} Also suppose that n μi ≤ 2.31 i If fi ∈ SHli α, b, 1 ≤ i ≤ n , then the integral operator Ik,n,l,μ class SHk α, b, Proof In Theorem 2.19, we consider λ Remark 2.21 If we set b F, defined by 1.27 , is in the 1 in Corollary 2.20, then we have 7, Theorem Theorem 2.22 Let α ≥ 0, b ∈ C − {0} and λ ≥ Also suppose that √ 2α √ 2−1 n μi < 2.32 i If fi ∈ SHli α, b, λ class Kk 0, b, λ ≤ i ≤ n , then the integral operator Ik,n,l,μ Proof Since fi ∈ SHli α, b, λ Re √ √ 2 b F, defined by 1.27 , is in the ≤ i ≤ n , by 1.22 we have Dli fi z −1 Dli fi z 2α √ 2−1 > 1 b Dli fi z −1 Dli fi z − 2α √ 2−1 2.33 Journal of Inequalities and Applications 13 for all z ∈ U Considering this inequality and 2.1 , we obtain √ λ z Dk f z b Dk f z √ n √ Dli fi z Re −1 μi b i Dli fi z 2Re √ √ 2− n n μi i √ √ 2− n μi Re μi −2α √ 1− √ √ n n μi 2−1 i √ 2α √ Dli fi z −1 Dli fi z b i i > √ μi Re √ √ b i n 2−1 Dli fi z −1 Dli fi z 2α √ 2−1 >0 μi i 2.34 for all z ∈ U Hence, by 1.8 , we have F ∈ Kk 0, b, λ Corollary 2.23 Let α ≥ and b ∈ C − {0} Also suppose that √ 2α √ 2−1 n μi < 2.35 i If fi ∈ SHli α, b, class Sk 0, b, ≤ i ≤ n , then the integral operator Ik,n,l,μ Proof In Theorem 2.22, we consider λ Remark 2.24 If we set b F, defined by 1.27 , is in the 1 in Corollary 2.23, then we have 7, Theorem References F M 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