Partial Sums of Starlike and Convex Functions

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 209, ARTICLE NO. AY Partial Sums of Starlie ad Covex Fuctios H. Silverma* Departmet of Mathematics, Uiersity of Charlesto, Charlesto, South Carolia Submitted by William F. Ames Received May 23, 1996 Let f z z2az be the sequece of partial sums of a fuctio f z z a z that is aalytic i z 2 1 ad either starlie of order or covex of order, 01. Whe the coefficiets a 4 are small, we deter- mie lower bouds o RefŽ z. f Žz 4, Ref Ž z. fž z.4, RefŽ z. f Ž z.4, ad Ref Ž z. fž z.4. I all cases, the results are sharp for each Academic Press 1. INTRODUCTION Let S deote the class of fuctios of the form f z z a z 1 2 that are aalytic ad uivalet i the uit dis z: z 14. A fuctio f i S is said to be starlie of order, 01, deoted S* Ž., if Rezff 4, z, ad is said to be coex of order, deoted KŽ., if Re1 zff 4, z. Let T* Ž. ad CŽ. be the subfamilies of S* ad KŽ., respectively, whose fuctios are of the form f z z a z, a * This wor was completed while the author was o sabbatical leave from the Uiversity of Charlesto as a Visitig Scholar at the Uiversity of Califoria, Sa Diego X97 $25.00 Copyright 1997 by Academic Press All rights of reproductio i ay form reserved.

2 222 H. SILVERMAN A sufficiet coditio for a fuctio of the form 1 to be i S* is that 2 a 1 Ž 3. ad to be i K is that 2 a 1. Ž 4. For fuctios of the form 2, these sufficiet coditios are also ecessary. See 3. I this ote, we will examie the ratio of a fuctio of the form 1 to its sequece of partial sums f z z2az whe the coefficiets of f are sufficietly small to satisfy either coditio 3 or 4. We will determie sharp lower bouds for Re fž z. f Ž z.4, Re f Ž z. fž z.4, RefŽ z. f Ž z.4, ad Ref Ž z. fž z.4. Sheil-Small 2 showed that if RefŽ z. f Ž z.4 for f K 0 occurs whe 1. I 1, a sharp lower boud was foud for RefŽ z. f Ž z.4 1 whe fkž.. E. M. Silvia 4 ivestigated lower bouds o RefŽ z. f Ž z.4 for ft* ad f CŽ.. She showed that RefŽ z. f Ž z.41ž 2. for f T* Ž. ad that RefŽ z. f Ž z.4ž 3. Ž 42. for f CŽ.. These results are sharp whe 1. Geerally, lower bouds o ratios lie Re fž z. f Ž z.4 or Ref Ž z. fž z.4 have bee foud to be sharp oly whe 1. I this paper, we determie sharpess for all values of. The lower bouds i questio are strictly icreasig fuctios of. I the sequel, we will mae frequet use of the well-ow result that Ž. Ž.4 Re 1 w z 1w z 0, z, if ad oly if w z 1cz satisfies the iequality wž z.z. Uless otherwise stated, we will assume that f is of the form 1 ad that its sequece of partial sums is deoted by f z z a z MAIN RESULTS THEOREM 1. If f of the form 1 satisfies coditio 3, the ReŽfŽ z. f Ž z.. Ž 1., z. The result is sharp for eery, Ž. 1 with extremal fuctio f z z 1 1 z.

3 STARLIKE AND CONVEX FUNCTIONS 223 Proof. We may write 1 fž z. 1 fž z az Ž Ž 1. Ž 1.. az 2 1 1AŽ z.. 1BŽ z. 1 a z 1 2 Set Ž1 Az. 1B Ž Ž z.. Ž1wŽ z.. Ž1wŽ z.., so that wž z. Ž Az BŽ z.. Ž2Az BŽ z... The Ž 1. Ž 1. a z 1 1 wž z a z Ž 1. Ž 1. a z ad wž z. Ž Ž 1. Ž 1.. a a Ž 1. Ž 1. a 2 1 Ž. Now w z 1 if ad oly if a a, which is equivalet to 2 1 a a 1. Ž It suffices to show that the LHS of 5 is bouded above by ŽŽ 2. Ž 1..a, which is equivalet to 1 1 ž / 1 ž 1 / 2 1 a a 0.

4 224 H. SILVERMAN Ž. 1 To see that f z z 1 1 z gives the sharp result, we observe for z re i that ž / fž z z 1 fž z. 1 1 whe r 1. 1 THEOREM 2. If f of the form 1 satisfies coditio 4, the ReŽfŽ z. f Ž z.. 21 Ž 1,z.The. result is sharp for eery, with extremal fuctio fž z. z ŽŽ 1. Ž 1.Ž z. Proof. We write Ž 1.Ž 1. fž z. Ž 2. 1 fž z. Ž 1.Ž 1. where az Ž Ž 1.Ž 1. Ž 1.. az 2 1 1wŽ z., 1wŽ z. 1 a z 1 2 Ž 1.Ž 1. Ž 1. a z 1 1 wž z a z Ž 1.Ž 1. Ž 1. a z Now Ž Ž 1.Ž 1. Ž 1.. a 1 wž z a Ž 1.Ž 1. Ž 1. a 2 1

5 STARLIKE AND CONVEX FUNCTIONS 225 if 2 Ž 1.Ž 1. a a 1. Ž The LHS of 6 is bouded above by Ž1.a if 1 1 ½ 2 2 Ž 1. a Ž 1.Ž 1. a 0, 1 5 ad the proof is complete. Remar. Our results i Theorems 1 ad 2 agree with those i 4 for the special case 1 ad are a improvemet whe 1. We ext determie bouds for f z f z. THEOREM 3. a If f of the form 1 satisfies coditio 3, the ReŽf Ž z. fž z.. Ž 1. Ž 22., z. Ž b. If f satisfies coditio Ž 4,. the ReŽ f Ž z. fž z.. Ž 1.Ž 1. ŽŽ 1.Ž 1. Ž 1... Ž. 1 Equality holds i a for f z z 1 1 z ad i Ž b. Ž. 1 for f z z z. Proof. write We prove a. The proof of b is similar ad will be omitted. We 21 fž z. 1 1 fž z az Ž Ž 1. Ž 1.. az 2 1 1wŽ z., 1wŽ z. 1 a z 1 2

6 226 H. SILVERMAN where Ž Ž 22. Ž 1.. a 1 wž z a Ž 1. a 2 This last iequality is equivalet to a a 1. Ž Sice the LHS of 7 is bouded above by ŽŽ. Ž 1..a 2, the proof is complete. We ext tur to ratios ivolvig derivatives. THEOREM 4. If f of the form 1 satisfies coditio 3,the for z, a ReŽfŽ z. f Ž z.. Ž 1., Ž b. ReŽf Ž z. fž z.. Ž 1. ŽŽ 1. Ž 1.Ž 1... I both cases, the extremal fuctio is fž z. z ŽŽ 1. Ž z. Proof. We prove oly a, which is similar i spirit to the proof of Theorem 1. The proof of Ž b. follows the patter of that i Theorem 3Ž a.. We write where 1 fž z. 1 wž z., Ž 1.Ž 1. f Ž z. 1 1wŽ z. Ž 1. Ž 1.Ž 1. a z 1 1 wž z a z Ž 1. Ž 1.Ž 1. a z 2 Now wž z.1if a a 1. Ž 8. Ž 1.Ž 1. 1.

7 STARLIKE AND CONVEX FUNCTIONS 227 Sice the LHS of 8 is bouded above by ŽŽ. Ž 1..a 2, the proof is complete. THEOREM 5. If f of the form 1 satisfies coditio 4,the for z, a ReŽfŽ z. f Ž z.. Ž 1., Ž b. ReŽf Ž z. fž z.. Ž 1. Ž 22.. I both cases, the extremal fuctio is fž z. z ŽŽ 1. Ž 1.Ž z. Proof. It is well ow that f KŽ. zf SŽ.. I particular, f satisfies coditio 4 if ad oly if zf satisfies coditio 3. Thus, a is a immediate cosequece of Theorem 1 ad Ž b. follows directly from Theorem 3 a. REFERENCES 1. L. Bricma, D. J. Hallebec, T. H. MacGregor, ad D. Wile, Covex hulls ad extreme poits of families of starlie ad covex mappigs, Tras. Amer. Math. Soc. 185 Ž 1973., T. Sheil-Small, A ote o partial sums of covex schlicht fuctios, Bull. Lodo Math. Soc. 2 Ž 1970., H. Silverma, Uivalet fuctios with egative coefficiets, Proc. Amer. Math. Soc. 51 Ž 1975., E. M. Silvia, O partial sums of covex fuctios of order, Housto J. Math. 11, No. 3 Ž 1985.,