On conditions for univalence of some integral operators

Transcription

1 Haceepe Journal of Mahemaics Saisics Volume 45 (2) (26), On condiions for univalence of some inegral operaors Daniel Breaz Virgil Pescar Keywords: Absrac In his paper, we obain new univalence condiions for he inegral operaors F G of analyic funcions defined in he open uni disk. Inegral operaor, univalence, ineger par, modulus. 2 AMS Classificaion: 3C45. Received : Acceped : Doi :.5672/HJMS Inroducion Le A denoe he class of funcions of he form f(z) = z + a 2z which are analyic in he open uni disc U = {z : z < }. Furher, by S we shall denoe he class of all funcions in A which are univalen in U. Pescar [7], has obained he following univalence crieria. Theorem. [7]Le γ C, f S, f(z) = z + a 2z If zf (z) f(z) zf(z), z U γ F γ(z) = max [ ( z 2 ) z z + a 2 + a 2 z ( ) γ f() d ], Deparmen of Mahemaics, Decembrie 98 Universiy of Alba Iulia, Faculy of Science, 59 Alba Iulia, Romania. dbreaz@uab.ro Deparmen of Mahemaics, Transilvania Universiy of Braşov, Faculy of Mahemaics Compuer Science, 59 Braşov, Romania. virgilpescar@unibv.ro

2 338 is in he class S..2 Theorem. [7]Le α, β, γ C, f S, f(z) = z + a 2z If zf (z) f(z) zf(z), z U, Reβ Reα > γ [ ], max z 2Reα z z + a 2 Reα + a 2 z G β,γ (z) = β is in he class S. β ( f() ) γ d We define he nex wo inegral operaors F = β ( ) α ( ) f() f[ δ ] () α[ δ ] d, where δ C, δ / [, ), α i C, f i A, i =, [ δ ], α α [ δ ] = δ ( ) α ( ) γ f() G [ γ ] (z) = γ γ f[ γ ] () α[ γ ] d, γ C, γ / [, ), α i C, f i A, i =, [ γ ], α α [ γ ] = γ. In his paper, we obain new univalence condiions for he inegral operaors F G. 2. Preliminary resuls In order o derive our main resuls, we have o recall here he following lemmas: 2. Lemma. [2]If he funcion f is regular in uni disk U, f(z) = z + a 2z ( z 2 ) zf (z) f (z), for all z U, he funcion f is univalen in U. 2.2 Lemma. [5]Le α be a complex number, Reα > f(z) = z + a 2z be a regular funcion in U. If z 2Reα Reα zf (z) f (z), for all z U, for any complex number β, Reβ Reα, he funcion F β (z) = β is in he class S. β f ()d β

3 Lemma. [3]If he funcion g is regular in U g(z) < in U, for all ξ U, he following inequaliies hold g(ξ) g(z) g(z)g(ξ) ξ z zξ (2.) g (z) g(z) 2 z 2, he equaliies hold in he case g(z) = ɛ z+u, where ɛ = u <. +uz 2.4 Remark. [3] For z =, from inequaliy (2.) we obain for every ξ U, g(ξ) g() g()g(ξ) ξ hence, g(ξ) ξ + g() + g()g(ξ). Considering g() = a ξ = z, g(z) for all z U. 3. Main resuls z + a + a z, 3. Theorem.Le M >, δ C, δ / [, ), α i C, for i =, [ δ ] α... α [ δ ] = δ. If f i A, f i(z) = z + a i 2z , for i =, [ δ ] zf i(z) f i(z) zf i(z), i =, [ δ ], z U, (3.) α α[ δ ] α α [ δ ] M, (3.2) α α [ δ ] [ ( M max z 2 ) ], (3.3) z z + c + c z where α a α [ δ ] a [ δ ] 2 c = M α α [ δ ], F = ( ) α ( ) f() f[ δ ] () α[ δ ] d is in he class S. Proof. We have f i A, for all i =, [ δ ] f i(z), for all i =, [ δ ]. ( ) z α ( Le g be he funcion g(z) = f (z) f[ δ ] ) (z) α[ δ ] z, z U. We have g() =. z Consider he funcion h(z) = M α α [ δ ], z U. (z) [ δ ]

4 34 The funcion h(z) has he form: Also, [ δ ] h(z) = M zf α α [ δ ] α i(z) f i(z) i. zf i(z) i= [ δ ] h() = M α α [ δ ] α ia i 2. i= By using he relaions (3.) (3.2) we obain ha h(z) < α a α [ δ ] a [ δ ] 2 h() = M α α [ δ ] = c. Applying Remark 2.4 for he funcion h we obain M α α [ δ ] ( z 2 ) z F z + c + c z, z U Consider he funcion H : [, ] R defined by We have H(x) = ( x 2 )x x + c + c x ; x = z. H M ( α α [ δ ] z 2 z + c ) z, z U.(3.4) + c z ( ) = c > max H(x) >. 2 + c x [,] Using his resul from (3.4) we have: ( z 2 ) z M [ ( α α [ δ ] max z 2 ) z z < Applying he condiion (3.3) in he form (3.5) we obain ha ( z 2 ) z, z U, from Lemma 2. we obain ha F [ δ ] S. ] z + c, z U.(3.5) + c z 3.2 Theorem. Le M >, γ, δ C, γ / [, ), α i C, for i =, [ γ ], α α n = [ γ ]. If f i A, f i(z) = z + a i 2z , for i =, [ γ ] zf i(z) f i(z) zf i(z), i =, [ γ ], z U, (3.6)

5 34 where α α [ γ ] α α [ γ ] M, (3.7) Reγ Reδ >, α α [ γ ] M max α a α [ γ ] a [ γ ] 2 c = M α α [ γ ], G [ γ ] (z) = γ is in he class S. Proof. We consider he funcion h(z) = Define he funcion p(z) = [ ( z 2 ) z z + c + c z ( ) α f() γ ( ) α ( ) f() f[ γ ] () α[ γ ] d. M α α [ γ ] h (z) h (z), z U. The funcion p(z) has he form: p(z) = [ γ ] M zf α α [ γ ] α i(z) f i(z) i. zf i(z) i= ], (3.8) ( ) f[ γ ] () α[ γ ] d By using he relaions (3.6) (3.7) we obain p(z) < α a α [ γ ] a [ γ ] 2 p() = M α α [ γ ] = c. Applying Remark 2.4 for he funcion h we obain M α α [ γ ] h (z) h (z) z + c + c z, z U z 2Reδ z h (z) Reδ h (z) M α α [ γ ] z 2Reδ z + c z, z U.(3.9) Reδ + c z Consider he funcion Q : [, ] R defined by Q(x) = x2reδ Reδ x x + c + c x ; x = z. We have Q ( ) 2 > max Q(x) >. x [,] γ

6 342 Using his resul in (3.9), we have: z 2Reδ Reδ zh (z) h (z) M [ z 2Reδ α α [ γ ] max z < Reδ Applying he condiion (3.8) in he relaion (3.), we obain ha z 2Reδ Reδ zh (z) h (z), z U from Lemma 2.2, we obain ha G [ γ ] S. References z ] z + c, z U.(3.) + c z [] D. Breaz, N. Breaz, Two inegral operaors, Sudia Universiais Babeş-Bolyai, Mahemaica, Cluj-Napoca, No.3, 22, 3-2. [2] J. Becker, Lownersche Differnialgleichung und quasikonform forsezbare schliche Funcionen, J. Reine Angew. Mah. 255(972), [3] G. M. Goluzin, Geomericeskaia eoria funkii kompleksnogo peremenogo, Moscova, 966. [4] Y.J. Kim, E.P. Merkes, On an inegral of powers of a spirallike funcion, Kyungpook Mah., J., 2 (972), 2, [5] N.N. Pascu, An improvemen of Becker s Univalence Crierion, Proceedings of he Commemoraive Session Simion Soilow, Universiy of Braşov, 987, [6] N.N. Pascu, V.Pescar, On he inegral operaors of Kim-Merkes Pfalzgraff, Mahemaica, Univ. Babeş-Bolyai, Cluj-Napoca, 32 (55), 2(99), [7] V. Pescar, On some inegral operaions which preserve he univalence, Journal of Mahemaics, Vol. XXX(997), pp. -, Punjab Universiy.