New Subclass of Multivalent Functions with Negative Coefficients inanalytic Topology

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1 AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN: EISSN: Journl home ge: New Subclss of Multivlent Functions with Negtive Coefficients inanlytic Toology Lieth Mjed Lieth A. Mjed, lecture, dertment of mthemtics College of Science,Diyl,Irq. Address For Corresondence: Lieth Mjed, Lieth A. Mjed, lecture, dertment of mthemtics College of Science, Diyl,Irq, E-mil: A R T I C L E I N F O Article history: Received 1 August 217 Acceted 1 October 217 Avilble online 18 October 217 A B S T R A C T In this resent er, we estblish new subclss of multivlent functions with negtive coefficients in unit disk = {z C: z < 1}. We obtin some roerties, like, theorem of coefficient inequlity, weighted men, subordintion theorems.ams subject clssifiction: 3C45. Keywords: Negtive coefficients, closure theorem,weighted men nd integrl oertor. Let be denote the clss of functions of the form: INTRODUCTION f z = z + n + z n+, N (1) which re nlytic nd n+- vlent in the oen unit disk U={z C: z < 1}. Let denote the subclss of of functions of the form: n + z n+, (, z ). (2) Note tht the uthors defined nd studied some clsses of nlytic functions like the form (1) in (Ibrhim, W., D. Mslin, 21) nd (Ans, A. nd D. Mslin, 215). For function f, let the Komtuoertor [6] defined by k δ c, f z = = z δ Г δ 1 z c tc 1 +n n+z n+ z log z t δ 1 f t dt (c >, δ > ) (3) We suose, n, c, μ, β denote the subclss of consisting of functions f which stisfy: ()( 1)z 2 k δ c, f z )ˊˊ μ kδ c, f z ˊˊ+2μ 1 z 2 < β, (4) Oen Access Journl Published BY AENSI Publiction 217 AENSI Publisher All rights reserved This work is licensed under the Cretive Commons Attribution Interntionl License (CC BY).htt://cretivecommons.org/licenses/by/4./ ToCite ThisArticle:Lieth Mjed., New Subclss of Multivlent Functions with Negtive Coefficients inanlytic Toology. Aust. J. Bsic & Al. Sci., 11(13): 2-24, 217

2 21 Lieth Mjed, 217 where δ >, < μ < 1, < β. nd k δ c, f is given by (3). Theorem (1): Let the function f be defined by (2).Then f, n, c, μ, β if nd only if n +, (5) +n where δ >, < μ < 1, < β. The result is shr for the function f z = z n + n + 1 βμ + 1 +n z n+, n 1. Assume tht the inequlity (5) holds true nd let z = 1,then from (4), we obtin ()( 1)z 2 (f z )ˊˊ γ μ f z ˊˊ + 2μ 1 z 2 = n + n n n +z n+ 2 γ 3μ 1 z 2 μ n + n + 1 n + n + 1 = +n n+z n+ 2 (6) + n 3 βμ 1 + βμ n + n n n + n + 1 (γμ + 1) n + by hyothesis. Hence by mximum modulus rincile, f, n, c, μ, β. Conversely, Letf, n, c, μ, β. Then ()( 1)z 2 (k δ c, f z )ˊˊ < β, z. μ k δ c, f z ˊˊ + 2μ 1 z 2 Tht is n+ (n+ 1) +n n + zn + 2 3μ 1 z 2 μ n+ n+ 1 +n n + zn + 2 3βμ ( 1), + n < β, (7) SinceRe(z) z for ll z z, we get Re n + (n+ 1) +n n + zn + 2 3μ 1 z 2 μ n + n+ 1 +n n + zn + 2 β, (8) we choose the vlue of z on the rel xis so tht (k δ c, f z )ˊˊ is rel. n + (n + 1) + n n +z n+ 2 3μγ 1 z 2 βμ n + n + 1 k=2 Letting z, through rel vlues, n + (n + 1) + n n + 3μβ βμ n + n + 1 we obtin inequlity (5). + n z n n,

3 22 Lieth Mjed, 217 Finlly, shrness follows, if we tke +n n+ n+ 1 βμ +1 z n+, n 1. (9) Corollry (1): Letf, n, c, μ, β. Then +n n+ n+ 1 (βμ +1), n 1. (1) In the following theorem, we obtin weighted men is in the clss, n, c, μ, β Definition (1)[5]: Let f 1 nd f 2 be in the clss, n, c, μ, β Then the weighted men of f 1 nd f 2 is given by: z = 1 2 j f 1 z j f 2 z, < j < 1. Theorem (2): Let f 1 nd f 2 be in the clss, n, c, μ, β Then the weighted menw j off 1 ndf 2 is lso in the clss, n, c, μ, β By Definition (1), we hve z = 1 2 = 1 2 j z,1 z n+ + (1 + j) z = z 1 2 j n +, j,2 z n+. j f 1 z j f 2 z,,2 z n+ Since f 1 nd f 2 re in the clss, n, c, μ, β so by Theorem (1), we get n + n + 1 βμ n k,1 μβ 1, nd Hence + n + n k,2 μβ 1. ] 1 2 j n +, j,2 = 1 2 ( j) + n n + n + 1 βμ + 1, j + n n + n + 1 βμ + 1,1 1 2 ( j)μβ j μγ 1 = μβ 1. 2 Therefore,, n, c, μ, β. In, Littlewood (1925) roved the following subordintion theorem. (see lso Duren 1983). Theorem (2) [4]: if fnd g re nlytic in U with f g then for > nd z = r e nd (< r < 1)

4 23 Lieth Mjed, 217 f z g(z) dθ (11) We will mke use the bove theorem to rove. Theorem (3): Let f, n, c, μ, β nd suose tht fis defined by +n n+ n+ 1 γμ +1 If there exists n nlytic function w given by w(z) n = +n n+ n+ 1 (γμ +1) μβ 1 then for z = r e nd (< r < 1) f r e z n+, n 1. (12) z n+, (13) f r e dθ, >. Let f(z)of the form (2) nd f n z defined by (12), then we must show tht k z k +n zk By lying Littlewood s subordintion theorem, it would suffice to show tht n + z n+ z n +. By setting +n dθ. We find tht n + z n+ = w z n+ = +n Which redily yieldsw =. Furthermore, by using (5), we obtin +n w z n+. z n+, w z n+ = +n 3μβ + ƞ + ƞ 1 n + z z n= +n n+ z n+ +n z < 1. Theorem (4): Let >. Iff, n, c, μ, β nd

5 24 Lieth Mjed, 217 then for z = r e nd (< r < 1), +n z n +, n 1 f r e r e f k dθ, (14) f z = z +n It is sufficient to show tht + n By setting n + hence n + f z = z (n + ) z n+ = w(z) n+ = k=2 n + n + 1 γμ + 1 z n+ Which redily yieldsw =. By using Theorem (1), we obtin w(z) n+ = z n= +n +n +n +n +n (n + ) z n + 1, z n+ 1, n 1. 3μγ 1 n + n + 1 (γμ + 1) 3μβ + ƞ + ƞ 1 k + + ƞ k + + ƞ 1 βμ + 1 μβ 1 n+ z n+ n + μβ 1 n + z n+ n + n + 1 βμ + 1 μβ 1 n + z n +. w z n+, z < 1. Conclusion: We obtin the roerties theorem of coefficient inequlity, closure theorem,weighted men nd integrl oertor. REFERENCES Ans,A.nd D. Mslin, 215.New subclss of -vlent functions with negtivecoefficients.,aip.c.p. Duren,P.L., Univlent Functions, Grundlehren der MthemtischenWisswnschften (Vol.259),Sringer-Verlg, New York. Ibrhim, W., D. Mslin,21. On certin clsses of multivlent nlytic functions., J. Mthemtics nd Sttistics., 6(3): Littlewood,L.E., 1925 On inequlities in the theory of functions, Prec. London Mth. Soc.,23: Miller,S.S.nd P.T. Mocnu,2. Differentil subordintions :Theory nd Alictions, Series on Monogrhs nd Text Books in Pure nd Alied Mthemtics (Vol. 225), Mrcel Dekker, New York nd Bsel. Slim,T.O., 21Aclss of multivlent function involving generlized liner oertor nd subordintion, Int. J. Oen Prob. Com. Anl., 2(2):