This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Some properties of an integral operator defined by convolution Journal of Inequalities and Applications 2012, 2012:13 doi:10.1186/1029-242X-2012-13 Muhammad Arif (marifmaths@awkum.edu.pk) Khalida Inayat Noor (khalidanoor@yahoo.com) Fazal Ghani (maxy20052001@yahoo.com) ISSN 1029-242X Article type Research Submission date 15 September 2011 Acceptance date 19 January 2012 Publication date 19 January 2012 Article URL http://www.journalofinequalitiesandapplications.com/content/2012/1/13 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1 Some properties of an integral operator defined by convolution Muhammad Arif *1 , Khalida Inayat Noor 2 and Fazal Ghani 1 1 Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan 2 Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan * Corresponding author: marifmaths@awkum.edu.pk E-mail addresses: KIN: khalidanoor@hotmail.com FG: univalentsfg@yahoo.com Abstract In this investigation, motivated from Breaz study, we introduce a new family of integral operator using famous convolution technique. We also apply this newly defined operator for investigating some interesting mapping properties of certain subclasses of analytic and univalent functions. 2010 Mathematics Subject Classification: 30C45; 30C10. Keywords: close-to-convex functions; convolution; integral operators. 2 1. Introduction Let A denote the class of analytic function satisfying the condition ( ) ( ) 0 0 1 0 f f ′ = − = in the open unit disc { } : 1 . z z = < U By * * , , , , S C S C and K we means the well-known subclasses of A which consist of univalent, convex, starlike, quasi-convex, and close-to-convex functions, respectively. The well-known Alexander-type relation holds between the classes * and , C S and * and , C K that is, ( ) ( ) * , f z C zf z S ′ ∈ ⇔ ∈ and ( ) ( ) * . f z C zf z K ′ ∈ ⇔ ∈ It was proved in [1] that a locally univalent function ( ) f z is close-to-convex, if and only if ( ) ( ) 2 1 Re 1 , , (1.1) i zf z d z re f z θ θ θ θ π ′′     + > − =   ′     ∫ for each ( ) 0,1 r ∈ and every pair 1 2 , θ θ with 1 2 0 2 . θ θ π ≤ < ≤ Let ( ) k P ξ be the class of functions ( ) p z analytic in U with ( ) 0 1 p = and ( ) 2 0 Re , , 2. 1 i p z d k z re k π θ ξ θ π ξ − ≤ = ≥ − ∫ This class was introduced in [2] and for 2, 0, k ξ = = the class ( ) k P ξ reduces to the class P of functions with positive real part. We consider the following classes: 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) : , : : , . k k k k zf z R f z A P z f z f z T f z A g z C P z g z ξ ξ ξ ξ ′     = ∈ ∈ ∈       ′     = ∈ ∃ ∈ ∈ ∈   ′     U U These classes were studied by Noor [3–5] and Padmanabhan and Parvatham [2]. Also it can easily be seen that ( ) * 2 0 R S = and ( ) 2 0 , T K = where * S and K are the well-known classes of starlike and close-to-convex functions. Using the same method as that of Kaplan [1], Noor [6] extend the result of Kaplan given in (1.1), and proved that a locally univalent function ( ) f z is in the class , k T if and only if ( ) ( ) 2 1 Re 1 , , (1.2) 2 i zf z k d z re f z θ θ θ θ π ′′     + > − =   ′     ∫ for each ( ) 0,1 r ∈ and every pair 1 2 , θ θ with 1 2 0 2 . θ θ π ≤ < ≤ For any two analytic functions ( ) ( ) ( ) 0 0 and g , , n n n n n n f z a z z b z z ∞ ∞ = = = = ∈ ∑ ∑ U the convolution (Hadamard product) of ( ) f z and ( ) g z is defined by ( ) ( ) ( ) 0 g , . n n n n f z z a b z z ∞ = ∗ = ∈ ∑ U Using the techniques from convolution theory many authors generalized Breaz operator in several directions, see [7, 8] for example. Here, we introduce a generalized integral operator ( ) ( ) , , : n n i i i I f g h z A A → as follows 4 ( )( ) ( ) ( ) ( ) ( ) ( ) 1 0 , , , 1.3 i i z n i n i i i i i i h t I f g h z f t g t dt t β α =     ′ = ∗           ∏ ∫ where ( ) ( ) ( ) , , i i i f z g z h z A ∈ with ( ) ( ) 0 i i f z g z ∗ ≠ and , 0 i i α β ≥ for 1, 2, , . i n = K The operator ( ) ( ) , , n i i i I f g h z reduces to many well-known integral operators by varying the parameters , i i α β and by choosing suitable functions instead of ( ) ( ) , . i i f z g z For example, (i) If we take ( ) ( ) for all 1 , 1 i z g z i n z = ≤ ≤ − we obtain the integral operator ( )( ) ( ) ( ) ( ) ( ) 1 0 , , 1.4 i i z n i n i i i i h t I f h z f t dt t β α =   ′ =       ∏ ∫ introduced in [9]. (ii) If we take 0 and 1 , i i n α = ≤ ≤ we obtain the integral ( )( ) ( ) 1 0 , i z n i n i i h t I h z dt t β =   =       ∏ ∫ introduced and studied by Breaz and Breaz [10]. (iii) If we take ( ) ( ) , 0, 1 i i z g z z β = = − we obtain the integral operator ( )( ) ( ) ( ) 1 0 , i z n n i i i I f z f t dt α = ′ = ∏ ∫ introduced and studied by Breaz et al. [11]. (iv) If we take 1 1 1, 0 and 1 n α β = = = in (1.4), we obtain the Alexander integral operator 5 ( )( ) ( ) 1 1 0 , z n h t I h z dt t   =       ∫ introduced in [12]. (v) If we take 1 1 1, 0 and n α β β = = = , we obtain the integral operator ( )( ) ( ) 1 1 0 , z n h t I h z dt t β   =       ∫ studied in [13]. In this article, we study the mapping properties of different subclasses of analytic and univalent functions under the integral operator given in (1.3). To prove our main results, we need the following lemmas. Lemma 1.1 [14]. Let ( ) ( ) k f z R ξ ∈ for 2, 0 1. k ξ ≥ ≤ < Then with 1 2 0 2 θ θ π ≤ < ≤ and , 1, i z re r θ = < ( ) ( ) ( ) 2 1 Re 1 1 . 2 zf z k d f z θ θ θ ξ π ′       > − − −           ∫ Lemma 1.2 [15]. If ( ) f z C ∈ and ( ) g z K ∈ , then ( ) ( ) g . f z z K ∗ ∈ 2. Main results Theorem 2.1. Let ( ) ( ) ( ) ( ) * * , and i i i k f z S g z C h z R ξ ∈ ∈ ∈ with 0 1, ξ ≤ < 2 k ≥ for all 1 . i n ≤ ≤ If ( ) ( ) 1 1 1 1, 2.1 2 n i i i k α ρ β =     + − − ≤         ∑ 6 then integral operator defined by (1.3) belongs to the class of close-to-convex functions. Proof. Let ( ) ( ) * * and . i i f z S g z C ∈ ∈ Then there exists ( ) i z C ϕ ∈ such that ( ) ( ) . i i f z z z ϕ ′ = Now consider ( ) ( ) ( ) ( ) ( ) ( ) * * * . i i i i i i f z g z z z g z z zg z ϕ ϕ ′ ′ = = Since ( ) * , i g z C ∈ then by Alexander-type relation ( ) . i zg z K ′ ∈ So, by Lemma 1.2, we have ( ) ( ) * , i i z zg z K ϕ ′ ∈ which implies that ( ) ( ) * i i f z g z K ∈ and hence, by using (1.1), ( ) ( ) ( ) ( ) ( ) ( ) 2 1 * Re 1 . (2.2) * i i i i z f z g z d f z g z θ θ θ π   ′′   + > −   ′     ∫ From (1.3), we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , , . 2.3 i i n i n i i i i i i h z I f g h z f z g z z β α =     ′ ′ = ∗           ∏ Differentiating (2.3) logarithmically, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n 1 1 , , 1 + 1 +1 , , n i i n i i i i i i i i i n i i i i i z f z g z I f g h z zh z h z I f g h z f z g z α β = = ′′ ′′   ′ ∗   + = −   ′ ′ ∗   ∑ ∑ 7 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n 1 1 1 1+ + 1 . n i i i i i i i i i i i i i z f z g z zh z h z f z g z α β α β = = =   ′′   ′ ∗     = + − +     ′   ∗     ∑ ∑ ∑ Taking real part and then integrating with respect to θ , we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 1 1 , , Re 1 Re 1+ , , n i i n i i i i i n i i i i i z f z g z I f g h z d d I f g h z f z g z θ θ θ θ θ α θ =   ′′   ′′ ∗     + =     ′ ′     ∗     ∑ ∫ ∫ ( ) ( ) ( ) ( ) 2 1 n n 2 1 1 1 Re 1 . i i i i i i i zh z d h z θ θ β θ α β θ θ = =   ′     + + − + −         ∑ ∑ ∫ Using (2.2) and Lemma 1.1, we have ( ) ( ) ( ) ( ) ( ) 2 1 1 , , Re 1 1 1 2 , , n n i i i i i i n i i i I f g h z k d I f g h z θ θ θ π α ρ β =   ′′       + > − + − −       ′         ∑ ∫ ( ) ( ) n 2 1 1 1 i i i α β θ θ =   + − + −     ∑ From (2.1), we can easily write ( ) ( ) 1 1 1 1 1. 2 n n i i i i i i k α β α ρ β = =     + < + − − ≤         ∑ ∑ This implies that ( ) 1 1 , n i i i α β = + < ∑ so, minimum is for 1 2 , θ θ = we obtain 8 ( ) ( ) ( ) ( ) 2 1 , , Re 1 , , , n i i i n i i i I f g h z d I f g h z θ θ θ π   ′′   + > −   ′     ∫ and this implies that ( ) ( ) , , . n i i i I f g h z K ∈ For 2 k = in Theorem 2.1, we obtain Corollary 2.3. Let ( ) ( ) ( ) ( ) * * * , and i i i f z S g z C h z S ξ ∈ ∈ ∈ with 0 1, ξ ≤ < for all 1 . i n ≤ ≤ If 1 1, n i i α = ≤ ∑ then ( ) ( ) , , . n i i i I f g h z K ∈ Theorem 2.4. Let ( ) ( ) and i k i k f z T h z R ∈ ∈ for 1 i n ≤ ≤ . If , 0 i i α β ≥ such that 0 i i α β + ≠ and ( ) ( ) 1 1, 2.4 2 n i i i i k α β β =   + − ≤     ∑ then ( ) ( ) , n i i I f h z defined by (1.4) belongs to the class of close-to-convex functions. Proof. From (1.4), we have ( ) ( ) ( ) ( ) ( ) ( ) 1 , . 2.5 i i n i n i i i i h z I f h z f z z β α =   ′ ′ =       ∏ Differentiating (2.5) logarithmically, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n 1 1 1 , 1 1+ + 1 . , n n i i i i i i i i i i i i i n i i I f h z z f z z h z h z f z I f h z α β α β = = = ′′     ′′ ′     + = + − +     ′ ′     ∑ ∑ ∑ Taking real part and then integrating with respect to θ , we get 9 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 1 1 n 1 1 , Re 1 Re 1+ Re , n n i i i i i i i i i i n i i I f h z z f z zh z d d d h z f z I f h z θ θ θ θ θ θ θ α θ β θ = =   ′′     ′′ ′       + = +       ′ ′       ∑ ∑ ∫ ∫ ∫ ( ) ( ) n 2 1 1 1 . i i i α β θ θ =   + − + −     ∑ ( ) ( ) n n 2 1 1 1 1 1 1 , 2 2 n i i i i i i i k k π α π β α β θ θ = = =     > − − − + − + −         ∑ ∑ ∑ where we have used Lemma 1.1 and (1.2) ( ) ( ) ( ) n 2 1 1 1 1 . 2 n i i i i i i i k α β β α β θ θ = =       = − + − + − + −             ∑ ∑ From (2.4), we can obtain ( ) 1 1. n i i i α β = + < ∑ So minimum is for 1 2 , θ θ = thus we have ( ) ( ) ( ) ( ) 2 1 , Re 1 . , n i i n i i I f h z d I f h z θ θ θ π   ′′   + > −   ′   ∫ This implies that ( ) ( ) , . n i i I f h z K ∈ For 2 k = in Theorem 2.4, we obtain the following result. Corollary 2.5. Let ( ) ( ) * , i i f z K h z S ∈ ∈ for 1 i n ≤ ≤ and ( ) 1 1, 2 n i i i i k α β β =   + − ≤     ∑ [...]... final manuscript Acknowledgments The authors would like to thank the reviewers and editor for improving the presentation of this article, and they also thank Dr Ihsan Ali, Vice Chancellor AWKUM, for providing excellent research facilities in AWKUM References [1] Kaplan, W: Close-to-convex Schlicht functions Michigan J Math 1, 169–185 (1952) 10 [2] Padmanabhan, K, Parvatham, R: Properties of a class of. .. with bounded boundary rotation Ann Polon Math 31, 311–323 (1975) [3] Noor, KI: On radii of convexity and starlikeness of some classes of analytic functions Int J Math Math Sci 14(4), 741–746 (1991) [4] Noor, KI: On some integral operators for certain families of analytic function Tamkang J Math 22, 113–117 (1991) [5] Noor, KI, Haq W, Arif M, Mustafa S: On bounded boundary and bounded radius rotations... generalization of close-to-convexity Int J Math Math Sci 15(2), 279– 290 (1992) [7] Bulut, S: A new general integral operator defined by Al-Oboudi differential operator J Inequal Appl 2009, Article ID 158408, 13 (2009) [8] Vijayvargy, L, Goswami, P, Malik, B: On some integral operators for certain classes of P-valent functions Int J Math Math Sci 2011, Article ID 783084, 10 (2011) [9] Frasin, BA: Order of convexity... of convexity and univalency of general integral operator J Franklin Inst 348(6), 1013–1019 (2011) [10] Breaz, D, Breaz, N: Two integral operator Studia Universitatis Babes-Bolyai, Mathematica, Clunj-Napoca 3, 13–19 (2002) [11] Breaz, D, Owa, S, Breaz, N: A new integral univalent operator Acta Univ Apulensis Math Inf 16, 11–16 (2008) 11 [12] Alexander, JW: Functions which map the interior of the unit... z ) defined by (1.4) belongs to the class of close-to-convex functions Competing interests The authors declare that they have no competing interests Authors’ contributions MA completed the main part of this article, KIN presented the ideas of this article, FG participated in some results of this article MA made the text file and all the communications regarding the manuscript All authors read and approved... of the unit circle upon simple regions Ann Math 17, 12–22 (1915–1916) [13] Miller, SS, Mocanu, PT, Read, MO: Starlike integral operators Pacific J Math 79, 157–168 (1978) [14] Noor, KI: On subclasses of close-to-convex functions of higher order Int J Math Math Sci 6(2), 327–334 (1983) [15] Ruscheweyh, S, Sheil-Small, T: Hademard product s of Schlicht functions and the Polya-Schoenberg conjecture Comment . upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Some properties of an integral operator defined by convolution Journal of Inequalities and Applications. (http://creativecommons.org/licenses /by/ 2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1 Some properties of an integral operator defined by. studied by Noor [3–5] and Padmanabhan and Parvatham [2]. Also it can easily be seen that ( ) * 2 0 R S = and ( ) 2 0 , T K = where * S and K are the well-known classes of starlike and close-to-convex