On Functions -Starlike with Respect to Symmetric Conjugate Points

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1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 201, ARTICLE NO O Fuctios -Starlike with Respect to Symmetric Cojugate Poits Mig-Po Che Istitute of Mathematics, Academia Siica, Nakag, Taipei 11529, Taiwa Zhwo-Re Wu Departmet of Applied Mathematics, Togji Uiersity, Shaghai , Chia ad Zhog-Zhu Zou Departmet of Mathematics, Huaihua Teacher s College, Huaihua, Hua , Chia Submitted by H. M. Sriastaa Received May 10, 1994 We itroduce here the otio of fuctios -starlike with respect to symmetric cojugate poits ad derive a covolutio theorem i this class. Moreover, a sharp coefficiet estimate ad a structural formula are give Academic Press, Ic. 1. INTRODUCTION Let A deote the class of fuctios fž. which are aalytic i the ope uit disk U : 14 ad ormalied by fž. 0 1 f Ž We refer to K, S, ad C, as usual, as the subclasses of A whose members are covex, starlike Ž w.r.t. origi. ad close-to-covex, respectively Žsee, e.g.,. 4. Let us deote by P the class of fuctios pž. which are regular i U ad satisfy the coditios pž. 0 1 ad RepŽ.4 0iU. * address: MAAPO@ccvax.siica.edu.tw X96 $18.00 Copyright 1996 by Academic Press, Ic. All rights of reproductio i ay form reserved.

2 26 CHEN, WU, AND ZOU We ow defie two operators D ad T as follows: Ž. 1 The Operator T. For f Ý a A, let Ý T fž. fž. fž. a Ž 1. a Ž. 2 The Operator D. For f A ad is a positive iteger, let D 0 fž. fž., Df Ž. f Ž., D 1 fž. DŽ D fž.., 1,2,3,.... It is easily see that T ad D are well-defied o followig properties: Ž. I Tad D are liear operators o A; Ž II. DT TD; Ž III. TT T. A ad have the R. Md. El-Ashwah ad D. K. Thomas 1 defied the class Sc of fuctios starlike with respect to symmetric cojugate poits as follows: Ž. DEFINITION 1 1. A fuctio f A with f f 0 for all U is said to be starlike with respect to symmetric cojugate poits, if it satisfies ½ 5 ½ 5 2 f Ž. Df Ž Re Re. 0, U. Ž 1. fž. fž. TfŽ. This class is deoted by S c, which may be viewed as the class of fuctios fž. starlike with respect to the fuctio f Ž. i the sese of K. Sakaguchi 3. I 1, some properties of this class are studied ad a structural formula is established. The geometric iterpretatio of the coditio Ž. 1 is that for every r i 0r1, the poit f Ž., re is i the left-had side of the directioal taget at the poit fž. of the image curve of the circle : r4 uder the fuctio fž.. I this paper, a ew class S Ž. of fuctios -starlike with respect to symmetric cojugate poits is defied ad some properties of this class such as a iterestig covolutio theorem, coefficiet estimates, ad a structural formula are obtaied. I additio, aother ew class C Ž. c of fuctios -close-to-covex with respect to symmetric cojugate poits is discussed.

3 STARLIKE FUNCTIONS 27 Ž. DEFINITION 2. A fuctio f A with f f 0iUis said to be -starlike with respect to symmetric cojugate poits, if it satisfies ½ 5 D D Ž 1. D 0 fž. Re 0, U Ž 2 0. D Ž 1. D TfŽ. for some 0. This class is deoted by S Ž. c. It is clearly see that S Ž. 0S. c c 0 Let us adopt the symbol D D 1 D, ad f Ž. D fž. ŽDŽ 1. D 0. fž. Ž 1. fž. f Ž.. We see that f S is equivalet to f Ž. S Ž. 0. c c 2. THE CLASS S AND HADAMARD PRODUCTS c I order to prove our mai results we eed the followig lemmas: LEMMA 1 2. If K, g S, ad p P, the ½ 5 ½ 5 Ž pg.ž. Ž. p Ž. g Ž Re Re. 0, U, Ž g.ž. Ž. gž. where, deotes, as usual, the Hadamard product of two fuctios fž. ad g i A, i.e., if f Ý2 a ad g Ý2b, the fg Ý a b. 2 LEMMA 2. If f S, the Tf S. c Proof. Settig we have p P. We also have Ž. Ž. From 3 ad 4, we obtai Df Ž. 2 f Ž. pž., Ž 3. Tf Ž. fž. fž. 2 f D f pž.. Ž 4. fž. fž. Tf Ž. ½ 5 DTf Ž. 1 Df Ž. D f Ž. 1 Ž pž. pž.., Tf Ž. 2 Tf Ž. 2 U. Ž 5.

4 28 CHEN, WU, AND ZOU Sice Re pž.4 0, for all U, ad Re pž. 4 RepŽ.4 0, for all U, therefore from Ž. 5 we have ½ 5 Tf Ž. Re 0, U. Tf Ž. This meas that Tf S. So the proof of Lemma 2 is complete. By Lemma 2, we see that if f S c, the f is a close-to-covex fuctio i the sese of Kapla, i.e., S c C; therefore f is uivalet i U. From Lemma 2, we immediately have THEOREM 1. Let 0. If f S, the D Tf S ad Tf S. c c Proof. Sice F S Ž. if ad oly if f D fždž c D fs, we ca see from Lemma 2 that Tf Ž. c TD f S. Further, by usig TD DT we get TD DT. Therefore, we have DTfŽ. TD f S. Moreover, TT T yields Ž. Ž. ½ Ž. 5 ½ 5 D D TfŽ. D D TfŽ. Re Re 0, U. D T Tf Ž. D Tf Ž. Hece Tf S. This completes the proof of Theorem 1. c Now, we prove a covolutio theorem for the class S, 0. c THEOREM 2. If f S Ž. c, 0, ad K with real coefficiets, the fs Ž.. c Proof. For f S, we have c DD f Ž. DTfŽ. pž. P. Ž 6. By Theorem 1, DTfS. Sice K with real coefficiets, oe ca easily verify that Ž. Ž. hold. From 6 ad 7 we obtai DTŽ f.ž. Ž. DTfŽ. DD Ž f.ž. Ž. DD f Ž. Ž 7. DDŽ f.ž. Ž. DD f Ž. Ž. pž. D fž.. DTŽ f.ž. Ž. DTfŽ. Ž. DTfŽ.

5 STARLIKE FUNCTIONS 29 By usig Lemma 1, we have ½ 5 DD Ž f.ž Re. 0, U; DTŽ f.ž. hece f S, which completes the proof of Theorem 2. c It is easy to verify that Ž 1. S Ž. c. So, from Theorem 2, we coclude that every covex mappig K with real coefficiets belogs to S c. But K S c, sice the fuctio Ž Ž4.i 1 e. is a covex fuctio but does ot belog to S c. COROLLARY 1. If f S,01, the Tf S. c Proof. The case 0 has bee cosidered i Lemma 2. We cosider ow the case 0 1. Let 1 t KŽ. Ý H dt, 1 Ž t 1 where It is well kow that K Ž. K ad has real coefficiets. Hece K f S Ž. c by Theorem 2. Theorem 1 yields the result DTŽK fž.. TfŽ. S. Ž. COROLLARY 2. If 0 1, the S S 0 S. c c c Proof. If f S Ž., the DTfŽ. S by Theorem 1. Sice c Df Ž. KŽ. DD f Ž. DTfŽ. DTf Ž. Tf Ž. K Ž. DTfŽ. where K Ž. is defied as i Corollary 1, the by usig Lemma 1, we have ½ 5 Df Ž Re. 0, U; Tf Ž. hece f S. This completes the proof of Corollary 2. c THEOREM 3. If f Ý a S Ž. 2 c, 0, the a, 2,3,.... Ž 8. 1 Ž 1. Ž. Especially, a 1 for f S 1 ad a for f S Ž. c c 0 S c. The estimate Ž. 8 is sharp ad the equality is attaied for the fuctio g Ž. gie by 1 Ž i. gž. Ý. 1 Ž 1. 2,

6 30 CHEN, WU, AND ZOU Proof. Sice fž. S c if ad oly if f Ž. D fž. 1 f f Ý2 1 1 a S c C, ad hece Ž1 Ž 1.. a, 2, 3, 4,.... This implies that the i- equality Ž. 8 holds. I order to prove that this estimate is sharp, we oly have to show that g S Ž., 0. This is equivalet to D Žg Ž.. c S c; therefore, it is sufficiet to prove that 1 Ž. DŽ gž.. Ý Ž i. S c. 2 Ž 1i. 2 Sice Ž. S ad TŽ. Ž., we have ½ 5 ½ 5 D Ž. Ž Re Re. 0, U. TŽ. Ž. This implies that Ž. S. Thus we complete the proof of Theorem 3. c Ž.. Remark. g 1i Ý 1 1 S Ž. 2 sc Ž 0.. THEOREM 4. A fuctio f S c if ad oly if there exist a fuctio p P ad a fuctio G S with real coefficiets such that G satisfies ad G Ž. 1 pž i. pž i., G Ž. 2 Ž 9. pž. GŽ i. f Ž. i. Ž 10. Proof. that Suppose that f S ; the there exists a fuctio p P such c pž. Tf Ž. fž.. Ž 11. Ž. By Lemma 2, Tf S, ad Lemma 1 yields G Tf 1 i S. It is easy to see that all the coefficiets of G are real. Moreover, we have From 11 ad 12, we obtai 10. Tf Ž. GŽ. igž i.. Ž 12. 1i

7 STARLIKE FUNCTIONS 31 We ow show that the fuctio G satisfies the coditio Ž. 9. By usig Ž. 5, after some computatios, we have Ž. G Ž. DGŽ. D TfŽ. Ž 1i. GŽ. GŽ. Tf Ž. Ž 1 i. DTf Ž. 1 Tf Ž. 1 i 1 1 Ž pž. pž.. 2 1i 1 Ž pž i. pž i... 2 Hece the coditio Ž. 9 holds. Coversely, for f A, if there exists a fuctio p P ad a fuctio GS with real coefficiets such that the coditios Ž. 9 ad Ž 10. hold, the we ca show that f S c. To do this, first we show that TfŽ. igž i.. From Ž 9. we get hece i GŽ i. G Ž i. Ž pž. pž.. ; 2 1 GŽ it. igž i. G Ž it. dt i pž. pž t. dt. Ž 13. From 10, we have H H 2 t 0 0 pž t. GŽ it. fž. ih dt. t Therefore after some easy computatios, we have 1 Tf Ž. fž. fž pž. pž t. GŽ it. ih dt 2 t 0 igž i. Ž from Ž Ž 14.

8 32 CHEN, WU, AND ZOU Next, from 10 we have Sice p P, from 14, we have ½ 5 ½ 5 pž. Tf Ž. f Ž.. Ž 15. Df Ž. f Ž Re Re. Re pž. 4 0, U. Tf Ž. Tf Ž. Therefore, we have f S, ad this completes the proof of Theorem 4. c Sice f ScŽ. if ad oly if D fs c, therefore from Theorem 4 we obtai the followig result immediately. THEOREM 5. A fuctio f S Ž. c, 0 if ad oly if there exist a fuctio p P ad G S with real coefficiets ad satisfyig the coditio Ž. 9 such that where f Ž. i pž t. GŽ it. t dt, Ž 16. H 1 0 Proof. Sice f S Ž. if ad oly if D ffs, from Ž 10. c c i Theorem 4, we have D fž. ipž. GŽ i.. Now, let f Ý a. The we have Hece 2 0 Ž. Ž. D f Df 1 D f Ž. f 1 f f Ž. 2 f Ž. Ý 1 Ž 1. a. 2 Ž. Ý 2 D f k a f,

9 STARLIKE FUNCTIONS 33 which implies which yields t Ž. H 0 f Ž. D fž. dt 1t 1 t ipž. GŽ i. H 0 dt 1t 1 1 i pž t. GŽ it. t dt, H 0 3. THE CLASS C c Ž. DEFINITION 3. A fuctio f A with f f 0iUis said to be -close-to-covex with respect to symmetric cojugate poits if it satisfies ½ 5 DD f Ž. Re 0, U Ž 17. DTŽ. for some 0 ad some S Ž.. This class is deoted by C Ž. c c. Especially, the class C Ž. 0 is deoted by C. c Obviously, S Ž. C Ž. c c for 0. I additio, we see that f C c if ad oly if D fc c. From Theorem 1 ad Ž 17., we ca prove COROLLARY 3. If f Ý a C Ž. 2 c, 0, the D f C, ad hece a, 2,3,4, Ž 1. This estimate is sharp, the extremal fuctio g Ž. beig gie Ž as i Theorem 3. by 1 Ž i. gž. Ý. 1 Ž 1. 2 By the same way as above, we ca prove: THEOREM 6. C Ž. C,01. c c THEOREM 7. If f C, 0, the Tf C ad D Tf C. c c c

10 34 CHEN, WU, AND ZOU THEOREM 8. If f C Ž. c, 0, ad k K with real coefficiets, the kfc Ž.. c By meas of Theorem 7 ad Theorem 8, we obtai COROLLARY 4. Let 0 1. If f C, the Tf C. c The proof is the same as i Corollary 1. Fially, we ca similarly prove THEOREM 9. A fuctio f C Ž. c if ad oly if there exist a fuctio p P ad a fuctio G S with real coefficiets such that ad 1 1 f Ž. i pž t. GŽ it. t dt, if 0, H 1 0 pž. GŽ i. f Ž. i, if 0. ACKNOWLEDGMENTS The authors express their heartfelt thaks to the referee for his critical review ad helpful suggestios for the improvemet of the paper. This work was supported, to the first author, by the Natioal Sciece Coucil of Taiwa uder Grat NSC M ; to the secod author, by the Natioal Natural Sciece Foudatio of Chia; ad to the third author, by Hua Provice Educatio Research Program Foudatio. REFERENCES 1. R. Md. El-Ashwah ad D. K. Thomas, Some subclasses of close-to-covex fuctios, J. Ramauja Math. Soc. 2 Ž 1987., St. Ruscheweyh ad T. Shiel-Small, Hadamard products of schlict fuctios ad Polya- Schoeberg cojecture, Commet. Math. Hel. 48 Ž 1973., K. Sakaguchi, A sufficiet coditio for uivalecy, i Topics i Uivalet Fuctios ad Its Applicatios, pp. 810, Srikaisekikehyusho ˆ ˆ Kokyuroku, ˆ ˆ Vol. 714, H. M. Srivastava ad S. Owa Ž Eds.., Curret Topics i Aalytic Fuctio Theory, World Scietific, Sigapore, 1992.