ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE


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1 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau Itroductio ad prelimiaries We deote by U r the disc { C : < r}, < r, U U Let A be the class of fuctios f which are aalytic i U ad f ( f ( Let S be the class of the fuctios f A which are uivalet i U Defiitio Let f ad g be two aalytic fuctios i U We say that f is subordiate to g, f p g, if there exists a fuctio ϕ aalytic i U, which satisfies ϕ(, ϕ( < ad f( g(ϕ( i U Defiitio A fuctio L : U x I C, I [o,, L(, t is a Loewer chai, or a subordiatio chai if L is aalytic ad uivalet i U for all U ad for all t, t I, t < t, L(, t p L(, t Lemma [] Let r (, ] ad let L(, t a (t +, a (t be aalytic i for all t I ad locally absolutely cotiuous i I, locally uiform with respect to U r U r For almost all t I suppose: L,t L,t ( ( p(,t, t U r where p is aalytic i U ad satisfies Re p(, t >, U, t I L(,t If a (t for t ad forms a ormal family i a ( t U r the for each t I, L has a aalytic ad uivalet extesio to the whole disc U I the theory of uivalet fuctios a iterestig problem is to fid those itegral operators, which preserve the uivalece, respectively certai classes of 7
2 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia uivalet fuctios The itegral operators which trasform the class S ito S are preseted i the theorems A, B ad C which follow The itegral operators studied by Kim ad Meres is that from the theorem: Theorem A [] If f S, the for C, belogs to the class S ( F ( du the fuctio F defied by: u A similar result, for other itegral operator has bee obtaied by Pfaltgraff i: Theorem B [3] If f S, the for C, belogs to the class S ( G ( [ f ' ( u ] du, the fuctio G defied by: A itegral operator differet of ( ad ( is obtaied by Silvia Moldoveau ad NN Pascu i the ext theorem: Theorem C [] If f S, the for C,, the fuctio I defied by: (3 I ( f ( udu belogs to the class S I this ote, usig the subordiatio chais method, we obtai sufficiet coditios for the regularity ad uivalece of the itegral operator: ( ( H ( d u where f A,,,, 7
3 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia Mai results Theorem Let f, f,, f A, C, < ad,,, R,,,, If: f ( (5 ( (, ( U,,, f ( the the fuctio H defied by ( is aalytic ad uivalet i U Proof Because ( f f ( f ( + a + is aalytic i U, there ( f f ( f ( exists a umber r (, ] such that for ay f( f( f U r The for the fuctio choose the uiform brach equal to at the origi, aalytic i (6 + b +, ad we have: ' ( U r : we ca f( f( f( g ( + b + e t (7 ( u g u du g(,t, where: t b ( + t (8 g (,t e + + e + + Let we cosider the fuctio: t t t ( t 3( (9 g,t f (,t + ( e e e g ( e Sice < we have g 3(,t e ( + e for ay t I ad it results that there exists r (, r ] such that g 3 (, t i U for all t I For the t t r 73
4 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia fuctio [g 3 (, t] / we ca choose a uiform brach, aalytic i U r for ay t I It results that the fuctio: where: ( L(, t [g 3 (, t] / [g (, t] /, t e ( ( g (,t du + t + ( e e ( f ( e f ( e is aalytic i U r Usig Lemma we will prove that L is a subordiatio chai We observe that L(, t a (t +, where: ( a (t e t ( + e t / Because < we have a (t for all t I ad t t t ad < t a ( t e ( e t + (3 [ ] t Re ( lim a ( t lim e t t if Re > or r r It follows that Let p : U r I, L(,t a ( t forms a ormal family of aalytic fuctios i L (,t p (,t L (,t t U r, I order to prove that p has a aalytic extesio with positive real part i U, for all t I it is sufficiet to prove that the fuctio: 7
5 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia (5 p (,t w (,t is aalytic i U for t I ad p (,t + (6 w (, t <, ( U, t I But w (,t < max w (,t iθ w ( e,t, θ R ad it is sufficiet that θ t (7 w ( e i,, ( t >, or (8 ( u ( u f'( u + f'( u + + f'( u u f '( u ( ( u, from (5 where u e t e iθ, u U, u e t It results (6 Hece the fuctio L is a subordiatio chai ad L(, t H ( from ( is aalytic ad uivalet i U Theorem If f, f,, f S,,, the for C,, the fuctio H defied by ( belogs to the class S Proof Because f S,, we have: the ad f '( f ( +, ( U, f '( ( ( + < f ( f '( ( ( ( + < f ( because The, from Theorem we obtai that H S 75
6 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia Example Let f (, f ( ad f 3 ( ( 5 The, for,, 3 we have: 3 3 f ( f3 ( ( ( 3 ( f + + ( ( + ( + ( H ( du ( For 3 H u ( 3 3 ( d 3 Refereces [] YJ, Kim, EP, Meres: O a itegral of powers of a spirallie fuctio Kyugpoo Math J, vol, r, (97, p 9 [] Silvia Moldoveau, NN, Pascu: Itegral operators which preserve the uivalece Mathematica (Cluj, 3 (55, r, (99, p [3] J Pfaltgraff: Uivalece of the itegral ( f '( c Bull Lodo Math Soc 7, (975, r 3, p 556 [] Ch, Pommeree: Uber die subordiactio aalytischer Fuctioe J Reie Agew Math 8 (965, p Authors: Maria E Gageoea ad Silvia Moldoveau  Trasilvaia Uiversity of Braşov Departamet of Mathematics Braşov 76
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