Classes of Uniformly Convex and Uniformly Starlike Functions as Dual Sets

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1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 16, ARTICLE NO. AY Classes of Uiformly Covex ad Uiformly Starlike Fuctios as Dual Sets I. R. Nezhmetdiov Faculty of Mechaics ad Mathematics, Kaza State Uiersity, 40008, Kaza, Russia Submitted by H. M. Sriastaa Received April 18, 1996 I this paper the classes of uiformly covex ad uiformly starlike fuctios are preseted as dual sets for certai fuctio families Ži the sese of covolutio theory.. The results are used to fid some sharp sufficiet coditios for fuctios, regular i the uit disk, to belog to the above classes Academic Press 1. INTRODUCTION Deote by A the class of ormalized fuctios f z zý a f z, regular i the uit disk E z : z 14. Cosider also its subclasses S, ST, CV, cosistig of the uivalet, starlike, ad covex fuctios, respectively. It is well kow that for ay f ST ot oly fž E. but the images of all circles cetered at 0 ad lyig i E are starlike with respect to 0. B. Pichuk posed a questio whether this property is still valid for circles cetered at other poits of E. A. W. Goodma 1 gave a egative aswer to this questio ad itroduced the class UST of uiformly starlike fuctios f ST such that for ay circular arc lyig i E ad havig the ceter at E the image fž. is starlike with respect to fž..a ecessary ad sufficiet coditio for f A to be uiformly starlike takes the form Re Ž z. fž z. Ž fž z. fž.. 0, z, E. Ž 1. I a similar way, the class UCV of uiformly covex fuctios is defied to iclude the fuctios f S such that ay circular arc lyig i E with the ceter E is carried by f ito a covex arc. A. W. Goodma 00-47X97 $5.00 Copyright 1997 by Academic Press All rights of reproductio i ay form reserved. 40

2 UNIFORMLY CONVEX FUNCTIONS 41 stated the criterio Re 1 Ž z. f Ž z. fž z. 0, z, E f UCV. Ž. Later F. Røig Ž ad idepedetly W. Ma ad D. Mida. 3 obtaied a more suitable form of the criterio, amely zf Ž z. fž z. Re 1 zf Ž z. fž z., z E f UCV. Ž 3. I the survey 3 it is observed that the study of the properties of UST is impeded by the absece of a simpler couterpart of the coditio Ž. 1. I the preset paper we deduce criteria for f to lie i UST or UCV stated i terms of dual sets Žsee. 4. For ay f, g A defie its covolutio Ž or Hadamard product. as f g z z Ý a f a g z. 4 We follow 4 to itroduce for ay set V A its dual set V * g A : Ž f g.ž z. z0, f V, z E 4. Ž 5. We show that both classes UST ad UCV are dual sets for certai families of fuctios from A. I particular, these results ca be used to obtai sharp forms of sufficiet coditios for f to be uiformly covex or starlike.. UNIFORMLY CONVEX FUNCTIONS THEOREM 1. Defie the fuctio family G g A : gž z. z1z 4zŽ i. Ž 1z. 3, where R 4. The UCV G*, ad i additio we hae a Ž g.ž 1. for all g G,. Proof. The coditio Ž. 3 is equivalet to the fact that the values of wui 1zf Ž z. fž z., z E, lie i the domai bouded by the parabola 1 u. Clearly, it ca be writte as 1 zf Ž z. fž z. Ž 1. i, z E, R. Hece, by the defiitio of the dual set, the first assertio of the theorem follows.

3 4 I. R. NEZHMETDINOV Ž For ay g G the coefficiets ca be writte as a g 1i. Ž i.. Cosider the expressio a g B t, Ž. Ž. where Bt t t 4t 4 4t 4,t 11. The the fuctio Bt Ž. decreases for 1 t ad icreases for t Ž. Ž., besides B 1 1 lim t Bt. Thus we have the required estimate which is attaied at 0 for all simultaeously. COROLLARY 1. Let f z z cz,. The fucv c 1Ž 1.. Proof. we get If f z zcz, where c 1Ž 1., the for ay g G 1 fg z z 1ca g z 1 ca g z 1 z 0, ad, by Theorem 1, f UCV. Coversely, suppose that f z zcz UCV. Cosider g z Ý 1 z G, so that Ž f g.ž z z 1 1c 1 z. Obviously, for c 1Ž 1. there is a E such that Ž f g.ž. 0, ad, cosequetly, f UCV. 0 Remark 1. I particular, whe, this implies Lemma from 3. COROLLARY. If f A satisfies the coditio Ý Ž 1. a Ž f. 1, Ž 6. the f UCV. The costat 1 o the right-had side is the best possible. Ž. The coditio 6 is verified directly by Theorem 1, whereas Corollary 1 implies the sharpess of the costat. Remark. This result was earlier obtaied i 5, Theorem 1, where the authors used quite a elemetary estimate ad verified the sharpess of the coditio for the fuctios with egative coefficiets. We ca also deduce a sufficiet coditio for f A to be i UCV of the form Ý a Ž f. 1, with the sharp costat 1. It is of iterest to compare this coditio with a well-kow covexity coditio Ý a Ž f.1 Žsee. 6.

4 UNIFORMLY CONVEX FUNCTIONS UNIFORMLY STARLIKE FUNCTIONS THEOREM. Let H hz A:hz z1 Ž iz1. Ž i.ž 1z.Ž 1z., where R, 1.The 4 UST H *. I additio, the uiform estimate C sup a Ž h.: h H4 d holds for all with the sharp costat d M , where M SŽ is the maximal alue of SŽ. Ž 1. 1 si Ž 1 si. si o 0. Here the extremal poit is the uique solutio 0 of the equatio 3 Ž cos cos 3. si 3 si 3 0, o the segmet LEMMA 1. Gie ay a, 0a1, the fuctio QŽ x. 1 x Ž 1 x. 4ax icreases o 0 x 1. To prove the lemma, just observe that for ay fixed x 0, 1 the derivative QŽ x. 1 Ž 1 x a. Ž 1 x. 4ax attais its miimum o 0 a 1at a1 ad, therefore, is oegative. LEMMA. The fuctio x 1 si x decreases o the iteral 0 x. The proof follows by usual differetiatio ad i view of the iequality 0 cos x x 1 si x, 0x. Let us write the criterio Ž. 1 of uiform starlike- Proof of Theorem. ess i the form Ž z. fž z. i Ž fž z. fž.., R, z, E Ž z.. As it is observed i 3, by the miimum priciple it is sufficiet to verify the coditio for z ; therefore, we may assume that z, 1. Hece from the defiitio of the dual set the first assertio immediately follows. Now, by settig expž i.,, we write až h. Ž A BC. Ž 1. FŽ., where A si si, B si si 1 si, C. Let us maximize F for all R. Note that lim F A for ay. I the same way it is easy to see that for B 0 we have

5 44 I. R. NEZHMETDINOV F. Otherwise, if B 0, the differetiatio yields FŽ. Ž B AC B 1., whece the maximal value of FŽ. o the whole lie is attaied for Ž 0 A C. D B, where D A C B, ad equals FŽ. Ž 0 A C D.. Thus, C max R Ž., where 0 R Ž. Ž 1. 1 Ž si si. ½ 5 1 si si 4 si cos 1 si. Note that R Ž Cosider the case 0 Ž 1.. The, settig Ž 1., we get 0 si si siž. si Ž cos si cot. Ž cos 1 si. 1 si, Ž 7. because cot 1,0, ad cos 1 si,0. Now by virtue of Lemma 1 ad Ž. 7 we obtai R Ž. SŽ. M max SŽ. 0. It remais to cosider the case whe Ž 1.. For Ž or 3. it is void. If 4, the by Lemma we have Hece 1 1 si 1 1 si 1. si si 1 si Ž 1. Ž 1. 1, ad, obviously, from R Ž. 1 Ž si si. we ifer that for Ž 1. the fuctio R Ž. 1.5 SŽ M, ad the proof of the estimate C M is complete. Now we study SŽ. for 0. Sice ½ 5 SŽ. Ž 1. 1 si Ž 1 si. si 1 si, the by Lemma we have S si M for 1.3. O the other had, writig 4 ½ 5 SŽ. Ž 1. 1 si Ž 1 si. 4 si,

6 UNIFORMLY CONVEX FUNCTIONS 45 from the iequality a b a b, where a, b 0, we get S 1 1 si. By a stadard calculus routie we establish that 1 si is a icreasig fuctio for 0 4, whece for it fol- 1 lows that S si M. We eed to make still arrower the iterval cotaiig the maximum poit of SŽ.. It ca be easily see that for by Lemmas 1 ad we have SŽ. Ž 1.½1 si Ž 1 si. 4 si cos 5. Now straightforward calculatios show that SŽ M, ; SŽ M, ; SŽ M, Thus the maximum of SŽ. is attaied o the iterval But from the equatio SŽ. 0 it follows that T Ž. 3 Ž cos cos 3. si 3 si 3 0. Let us prove that the fuctio TŽ. has the uique root o the iterval We compute TŽ , T Ž , ad TŽ , T Ž We show that T Ž. 0 o 0.8; 1.3 which implies the uiqueess of the root of the equatio alogside the covergece of the Newto method with the iitial approximatio 1.3. To this ed, let us cosider the followig three cases. The secod derivative equals T Ž. 3 Ž cos 9 cos 3. 3 Ž si 3 si 3. 6Ž cos cos 3. si 3. Ž. a For the fuctios cos, cos 3, ad si 3 decrease, while si icreases. I particular, we have Ž cos 9 cos 3. Ž cos cos , whece 3 Ž cos 9 cos Ž cos cos Similarly, Ž 6 si 9 si Ž 6 si si ; 6Ž cos cos Ž cos 0.9 cos ; si 3 si Thus, T Ž for

7 46 I. R. NEZHMETDINOV b Let Reasoig as above, we obtai T Ž Ž cos cos Ž 6 si si Ž cos 1.1 cos.7. si Ž. c If , the T Ž Ž cos cos Ž 6 si si Ž cos 1.3 cos 3.9. si This completes the proof of the theorem. Now we apply this result to determie the greatest value of such that the coditio Ý a Ž f. Ž 8. implies that f UST. A. W. Goodma showed i 1 that for this is valid; however, for the sufficiecy of Ž. 8, must ot exceed Fially, we have the followig COROLLARY 3. The coditio Ž. 8 implies the uiform starlikeess of f for 1 0 M , where M is determied i Theorem. If 0, the there exists a fuctio f UST satisfyig Ž. 8. Ž. Proof. Set i 8. The the estimate 0 1 fh z z 1 Ý a f a h z 1z M Ý a f, valid for ay h H, ad Theorem imply that f UST. Now, let 1 M with M SŽ.. Note that R Ž. SŽ as. Therefore, a positive iteger N ca be chose such that R Ž N. 1. Put also Ž A C D. N 0 0 B, where the expressios A, B, C, D are defied as i the proof of Theorem. Hece for a 1 certai fuctio h0 H the estimate an h0 N holds. Cose- 1 N quetly, the fuctio f z zn z satisfies the coditio Ž. 0 8 ; 1 N1 however, f0 h0 z z1n an h0 z vaishes at some poit E,so fust. Remark 3. Ulike its couterpart from Theorem 1, the estimate C d is attaied for o. We may cosider specific sharp estimates C d, where d max R Ž.. We state the followig 0

8 UNIFORMLY CONVEX FUNCTIONS 47 COROLLARY 4. Let f z z cz,. The f UST c 1d, with d defied as aboe. For we have d ad the corollary gives a result from 1. O the other had, computatios show that d , d , so d seems to be a icreasig sequece; however, this is still to be proved. ACKNOWLEDGMENT I express my gratitude to the referee for valuable suggestios ad commets. REFERENCES 1. A. W. Goodma, O uiformly starlike fuctios, J. Math. Aal. Appl. 155 Ž 1991., A. W. Goodma, O uiformly covex fuctios, A. Polo. Math. 56, No. 1 Ž 1991., F. Røig, A survey o uiformly covex ad uiformly starlike fuctios, A. Ui. Mariae Curie-Skłodowska Sect. A 47 Ž 1993., St. Ruscheweyh, Duality for Hadamard products with applicatios to extremal problems, Tras. Amer. Math. Soc. 10 Ž 1975., K. G. Subramaia, G. Murugusudaramoorthy, P. Balasubrahmayam, ad H. Silverma, Subclasses of uiformly covex ad uiformly starlike fuctios, Math. Japo. 4, No. 3 Ž 1995., A. W. Goodma, Uivalet fuctios ad oaalytic curves, Proc. Amer. Math. Soc. 8 Ž 1957.,