Proceedings of an International Conference On
New Trends •
1n Geometric Function Theory and Applications Editors: R. P...
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Proceedings of an International Conference On
New Trends •
1n Geometric Function Theory and Applications Editors: R. PARVATHAM & S. PONNUSAMY
World Scientific
Proceedings of an International Conference On
New Trends • 1n Geometric Function Theory and Applications In Honour of Professor K. S. PADMANABHAN
Editors: R. PARVATHAM & S. PONNUSAMY
~~hworld Scientific
~Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Farrer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07~ UK office: 73 Lynton Mead, Toueridge, London N20 8DH
NEW TRENDS IN GEOMETRIC FUNCTION THEORY AND APPLICATIONS Copyright © 1991 by World Scientific Publishing Co. Pte. Ltd. All righls reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without wrillen permission from the Publisher.
ISBN 981-02-0482-5
Printed in Singapore by JBW Printers & Binders Pte. Ltd.
The International Conference on New Trends in Geometric Function Theory and applications was held under the auspices of the University of Madras, Madras, India during July 26-29, 1990. This conference, organized under the Chairmanship ofProfessor A. Gnanam, Vice-Chancellor, University ofMadras, was inaugurated by the Honourable Minister, Professor M.G.K. Menon, Minister for Science and Technology, Govt. ofIndia. This International conference, the first of its kind in Mathematics in the 133 years old University ofMadras was organized by the Function Theorists who worked with Professor K.S. Padmanabhan, Director, Ramanujan Institute for Advanced Study in Mathematics, University of Madras. The Conference provided an opportunity for his students, grand students, friends and colleagues to honour him on his completion of sixty years of age. Over one hundred Mathematicians from several renowned institutions of India and of sixteen other countries, participated in the conference. The articles in the proceedings are associated with lectures delivered and papers presented at the conference. A session on open problems was also held during the conference and a list of such problems is included in this volume. The Editors would like to thank the many people who cooperated to make the conference a success and this volume possible. We express our appreciation and thanks to the World Scientific Publishing Co., (PVT) Ltd., in bringing out this volume. We also thank Shri N. Swaminathan, CoMPUPRINT, 51 Giri Road, Madras 600 017 and the City Printing Works for excellent typing of the manuscripts.
Rajagopalan PARVATIIAM, Ramanujan Institute, University of Madras, MADRAS
Saminathan PONNUSAMY, SPIC Science Foundation, MADRAS.
Madras, February 1991.
v
Professor K.S. PADMANABHAN, M.A., M.Sc., Ph.D. The organising committee of the International Conference on New Trends in Geometric Function Theory and applications is proud to dedicate the Proceedings of the International Conference on New Trends in Geometric Function Theory and applications to Professor K.S. Padmanabhan on his completion of sixty years of age. Padmanabhan was born at Palayamkottai- the citadel of Education in the Southern region ofTamil Nadu on March 19, 1930- the only son of Sri. P. Srinivasan and Shrimati Pankajaammal Srinivasan. Having lost his father in the early childhood, Padmanabhan was brought up with great care and affection by his mother. He was educated at Tirthapati High School, Ambasamudram -a beautiful serene town on the banks of the river Thambrabarani. Then he entered St. Xavier's College, Palayamkottai (affiliated to University of Madras in those days) for a basic degree in Mathematics. Later he went to St. Joseph's College, Tiruchy (also affiliated to University of Madras in those days) to persue his higher studies in Mathematics. He obtained a Master's degree in Mathematics in the year 1951 from the University of Madras. Soon after getting the Master's degree he moved to Vijayawada- a town on the banks ofthe river Krishna- to serve as lecturer in the S.R.R. & C.V.R. College. He joined the faculty of Annamalai University in 1952 and served there till 1969. While he was at Annamalai University, he first obtained the M.Sc., degree by research. Later he had taken up research under the able guidance of Professor V.G. lyer. He received the Ph.D., degree from the Annamalai University in 1966 for his work on Univalent functions. In 1969 Professor C.T. Rajagopal, then Director ofRamanujan Institute for Advanced Study in Mathematics, University of Madras, brought Padmanabhan to the Institute to strengthen the School of Analysis in the Institute. He served the University ofMadras in the capacity of a Reader upto 1975 and then as a Professor. Twice he held the directorship ofthe Ramanujan Institute forAdvanced Study in Mathematics. (1978-80 and 1987-90). His tenure of directorship can be described as the Golden Era for the Ramanujan Institute for Advanced Study in Mathematics. Under his able directorship the Institute had flourished with fabulous grants and was bubbling with academic activities. He has revived many schemes such as COSIST, during his directorship. Padmanabhan's mathematical interests lie mainly on Univalent Functions, Riemann Surfaces and Quasi Conformal Mappings and his out put of research is shown in his papers listed in this volume. In his chosen area of Analytic Function Theory, he has made significant contributions. He had guided numerous students for M.Phil., degree and twelve students had taken Ph.D., degrees under his supervision. He is one ofthe founder members ofthe Ramanujan Mathematical Society and the Editor-in-Chiefofthe Journal of Ramanujan Mathematical Society right from its installation till date. He is a scholar in the Sanskrit language. Padmanabhan's family life is a happy one. In 1955 he married Lakshmi, daughter of another mathematician popularly known as Calculus Srinivasan. Padmanabhans have two daughters and two sons, all well settled in life. R. PARVATHAM
·----------------------------------'
vi
Prof. K.S. Padmanabhan
vii
1. On generalised Taylor expansion, Math. Student (1957).
2. On the radius ofunivalence and starlikeness fora certain Analytic functions I, J. Indian Math. Soc. 29 (1965), 71-80. 3. On the radius ofunivalence and starlikeness for certain Analytic functions II, J. Indian Math. Soc. 20 (1965). 4. On certain classes ofmeromorphic functions in the unit circle, Math. Z. 89 (1965), 98-107. 5. On the radius of univalence and starlikeness for a certain class of meromorphic functions, J. Indian Math. Soc. 30 (1966), 202- 212. 6. On the radius of convexity of a certain class of meromorphically starlike functions in the unit circle, Math. Z. 91 (1966), 308- 313.
7. Coefficient estimates for a certain class ofmeromorphic starlike multivalent functions, J. Lond. Math. Soc. 42 (1967). 8. On certain classes of starlike functions in the unit disc, J. Indian Math. Soc. 32 (1968), 89-103. 9. Estimates of growth for certain convex and close to convex functions in the unit disc, J. Indian Math. Soc. 33 (1969), 37 - 48. 10. On the radius ofunivalence ofcertain classes ofanalytic functions, J. Lon d. Math. Soc. (2), 1 (1969), 225-231. 11. On a certain class of functions whose derivatives have a positive real part in the unit disc, Annales Polon. Math. 23 (1970), 73- 81. 12. On the partial sums ofcertain analytic functions in the unit disc, Ann ales Pol on. Math. 23 (1970), 85- 92. 13. The radius of univalence and starlikeness of a certain class of analytic functions, Annales Polon. Math. 26 (1972), 147- 157. 14. (with R. Parvatham), On the univalence and convexity of partial sums of a certain class ofanalytic functions whose derivatives have positive real part, Indian J. Math. 16 (1974), 66-77. 15. (with R. Parvatham), Radii of starlikeness of certain classes of Analytic functions, Mathematica (Cluj) 16 (1974), 143- 157. 16. On the arithmetic mean of univalent convex functions, Glas. Mat. 9 (1975), G5- 68. 17. (with R. Parvatham), Radius of convexity of partial sums of a certain power series, Indian J. Math. 17 (1975), 133 -138. 18. (with R. Parvatham), Properties of a class of functions with bounded boundary rotation, Annales Polon. Math. 31 (1975), 311 - 323. 19. (with R. Parvatham), On functions with bounded boundary rotation, Indian J. Pure Appl. Math. 6 (1975),1236- 1247.
viii
,.--------------------------------------
'
20. (with K.A. Narayanan), A brief introduction to Nevanlinna theory of meromorphic functions, Hyperbolic Complex Analysis (Proc. All India Sem., Ramanujan Inst., 1977), 1 - 20. 21. (with R. Parvatham), Radii of starlikeness of certain classes of Analytic Functions, Rev. Roum. Math. Pures et appl. 23 (1978), 1545- 1551. 22. (with R. Parvatham), On a certain class offunctions with bounded boundary rotation, Rev. Roum. Math. Pures et appl. 21 (1978),1077 -1084. 23. (with M.A. Nasr), On p-valent BazileviC and Alpha convex functions, J. Madras University B. 41, (1978), 41- 45. 24. (with J. Thangamani), On Alpha-starlike and Alpha close-to-convex functions with respect to symmetric points, J. Madras University, 42 (1979), 8- 11. 25. Developments in the theory of univalent functions, Symposia on Theoretical Physics and Math. Vol. 8 Planum Press, New York. 26. (with R. Parvatham), On certain generalised close-to-star functions in the unit disc, Annales Polon. Math. 37 (1980), 1 - 11. 27. (with R. Bharati), On a Close-to-Convex Functions II, Glas. Mat. 16 (36) (1981), 235-244. 28. (with J. Thangamani),The effect ofcertain integral operators on subclasses ofstarlike functions with respect to symmetric points, Bull. de Ia Soc. Sci. Math. de Ia R. S.de Roumanie 25 (1982), 355-360. 29. (with G.L. Reddy), On analytic functions with reference to the Bernardi integral operator, Bull. Austral. Math. Soc. 25 (1982), 387- 396. 30. (with R. Bharati),On a subclass of univalent functions I, Ann ales Pol on. Math. 43 (1983), 57- 64. 31. (with R. Bh~ati),On a subclass of univalent functions II, Annales Polon. Math. 43 (1983), 73- 78. 32. (with M.R. Rangarajan)OnAl-Amiri analogy of Alpha-Convex Functions, J. Indian Math. Soc. 47 (1983), 223-229. 33. (with M.S. Ganesan), A radius of Convexity Problem, Bull. Austral. Math. Soc. 28 (1983), 433-439. 34. (with G.L. Reddy), p-valent regular functions with negative coefficients, Comment. Math. Univ. St. Paul 32 (1983), 39- 49. 35. (with G.L. Reddy), Linear combinations of regular functions with negative coeffi· cients, Pub!. Inst. Math. (Beogard) 34 (48) (1983), 109 -116. 36. (with M.S. Ganesan), Convolution conditions for certain classes ofanalytic functions, Indian J. Pure Appl. Math. 16 (7) (1984), 777- 780. 37. (with G.L.Reddy),Somepropertiesoffractional integrals and derivativesofunivalent functions, Indian J. Pure Appl. Math. 16 (1985), 291 - 302. 38. (with G.L. Reddy), Distortion theorems for fractional integrals and derivatives, J. Madras Univ. Sec. B 48 (1985), 70- 92. ----- ___!
ix
39. (with R. Parvatham), Some applications ofdifferential subordination, Bull. Austral. Math. Soc. 32 (1985), 321- 330. 40. On the History of Bieberbach's conjecture and its recent solution, Math. Student 43 (1985), 209-216. 41. (with R. Manjini), Certain classes of analytic functions with negative coeffocients, Indian J. Pure Appl. Math. 17 (10) (1986), 1210-1223. 42. (with R. Manjini), Certain applications ofdifferential subordination, Publications de L' Institute Mathematique, 39 (53) (1986), 107 -118. 43. (with R. Parvatham), On Analytic functions and differential subordination, Bull. Math. de Ia Soc. Sci. Math. de Ia R.S. de Roumanie, 31 (79) (1987), 237-248. 44. (with R. Manjini), On generalization of pre-starlike functions with negative coefficients, Bulletin of the Institute of Mathematics, Academia Sinica 15 (1987), 329-343. 45. (with Sampath Kumar), On a subclass of Lowner chains, Jour. Math. Phy. Sci. 21 (1987), 653- 664. 46. (with R. Manjini), Certain classes of analytic functions with negative coef{ocients II, Indian J. Pure Appl. Math. 18 (2) (1987), 159- 172. 47. (with R. Manjini), Convolutions of pre-starlike functions with negative coefficients, Annales Univ. Mariae Curie- Sklodowska 41 (1987). 48. (with R. Parvatham and T. N. Shanmugam), a-convexity and a-close-to-convexity preserving integral operators, Annales Univ. Mariae Curie- SlsJ'odowska, 41 (1987), 89-97. 49. (with R. Manjini), Differential subordination and meromorphic functions, A.nnales Polon. Math. 49 (1988), 21 - 30. 50. (with R. Parvatham and S. Radha), On a Mocanu type generalization of the Kaplan class K (a, /3) of analytic functions, Ann ales Polon. Math. 49 (1988), 35 - 44. 51. (with R. Manjini), Subclasses of meromorphic univalent functions with positive coef{I.Cients, Bull. Math. de Ia Soc. Math. de Ia R.S. de Roumanie, 32 (1988), 327 - 340. 52. (withM.S. Ganesan), Convolutions ofcertain classes ofunivalent functions with nega· tive coefficients, Indian J. Pure Appl. Math. 19 (9) 880-889 (1988). 53. (with M. Jayamala), On p-valent analytic functions with reference to Bernardi and Ruscheweyh integral operators, Publ. Inst. Math. (Beograd) 46 (1989), 86 - 90. 54. (with M. Jayamala), Fixed coef{I.Cients for subclasses of starlike functions, Indian J. Pure Appl. Math. 21 (1990), 366- 378.
X
International Conference on
New Trends in Geometric Function Theory and
Applications 26 - 29 July 1990
Ramanujan Institute for Advanced Study in Mathematics University of Madras Madras • 600 005, INDIA.
Chairman
Prof. Dr. A Gnanam, Vice-Chancellor, University of Madras, Madras- 600 005.
Vice-Chairman
Shri. A V. Shetty, I.A.S., Registrar, University of Madras, Madras - 600 005.
Convener
Dr. R. Parvatham, Ramanujan Institute, University of Madras, Madras - 600 005.
-------------------·
xi
The Chairman and the Convenor ·
xii
1. University of Madras 2. University ofHyderabad 3. Indian Institute of Technology, Kanpur 4. University Grants Commission , New Delhi 5. Department of Science and Technology, New Delhi 6. National Board for Higher Mathematics, Bombay 7. Department of Atomic Energy, Bombay 8. Council for Scientific and Industrial Research, New Delhi 9. Indian National Science Academy, New Delhi 10. Ministry of Defence, New Delhi 11. Tamil Nadu Academy of Sciences 12. UNESCO, New Delhi 13. COSTED, Madras 14. British Council; Madras 15. I.M.U.C.D.E.
xiii
Honourable Minister Prof. M.G.K. Menon inaugurates the conference xiv
[Academic Programme J July 26,1990
Session 1 10.15- 11.30
Inaugural Ceremony
Invocation Welcome by
Shri. Ashok Vardhan Shetty, I.A.S., (Vice-Chairman) Registrar, University of Madras.
Presidential Address by
Prof. A. Gnanam (Chairman) Vice-Chancellor, University of Madras.
Inaugural Address by
Honourable Minister Prof. M.G.K. Menon, Minister for Science and Technology, Govt. of India.
Release Souvenir by
Shri. M. Gopalakrishnan, Chairman of Indian Bank.
First Copy Presented to
Prof. Iqbal Unnisa, Directress, Ramanujan Institute.
Unveiling the Portrait of KS. Padmanabhan
Prof. A. Gnanam, Vice-Chancellor, University of Madras.
Felicitations by
Prof. C.S. Seshadri, F.R.S., Prof. M.S. Rangachari.
Vote of Thanks by
Dr. R. Parvatham (Convener). NATIONAL ANTHEM
Session 2 112.00- 13.00
Keynote Address
Prof. S.S. Miller (U.S.A) Classes of Univalent Integral Operators.
XV
Session 3
Chairman :Prof. S.S. Miller (U.S.A)
14.00- 15.50
Pro(. J.M. Anden~on (U.K) The boundary behaviour of Bloch functions and Univalent functions. Prof. I. V. Anandam (Saudi Arabia) Funtions of Generalised bounded argument rotation. Prof. S. Owa (Japan) On certain subclasses of analytic functions. Prof. M.A. Nasr (Egypt) On Fekete-Skego Problem for close-to-convex functions of order p.
July 27, 1990
Session 1
Chairman: Prof. J.M. Anden~on (U.K)
9.00-1 0.30
Prof. D.K. Thomas (U.K) Bazilevic functions with logarithmic growth Prof. St. Ruscheweyh (West Germany) Convolutions in Geometric Function Theory - Open problems and recent results. Prof. S. Nag (India) String theory and the Teichmiiller Space of Riemann Surfaces.
Session 2 10.45- 13.00
Discussion on Status Report of NCMER sponsored by DEPARTMENT oF SciENCE AND TEcHNOLOGY,
Govt. oflndia.
Dr. S.K Gupta Prof. M.S. Rangachari Prof. St. Ruscheweyh
Pro(. D.K Sinha Prof. J.M. Anderson Dr. R. Parvatham
General Discussion
xvi
SessionS
Chainnan :Prof. S. Owa (Japan)
14.00- 16.00
Prof. V. Singh (India) Some inequalities for starlike and spiral-like functions. Prof. R. Nakki (Finland) John disks. Prof. R. W. Barnard (U.S.A) Problems in Geometric Function Theory. Prof. Rashian Md. Ali (Malaysia) Marty transformation determined by a coefficient inequality.
Session4
Chainnan :Prof. O.P. Junl!ja (India)
16.25- 17.55
Dr.S.Radha One generalised Pascu class of functions. Dr. Rajalakshmi Rajagopal Meromorphic functions and differential subordination. Dr. V. Srinivas Extreme points and Support points of certain class of analytic functions. Dr. M.S. Kasi On a class ofmeromorphic Univalent functions with negative coeffiCients. Dr. S.S. Bhoosnurmath Modifred Hadamard product of certain analytic functions. Dr. Vinod Kumar Starlikeness of an integral operator. Mr. Ashok Kumar Certain class of pre-starlike functions.
xvii
July 28, 199!1'
Session 1
Chainnan :Prof. St. Ruscheweyh (West Gennany)
9.00-10.30
Prof. P. Singh (India) N-subordination and N-symmetric points. Prof. N.S. Sohi (India) An application of fractional calculus for a class of analytic functions. Prof. B.A. Uralegaddi (India) Certain subclasses of analytic functions. Prof. E.M. Silvia (U.S.A) Inclusion relations between classes of functions defined by subordination.
Session2
Chainnan :Prof. D.K. Thomas (U.K)
10.45- 11.45
Prof. J. Stankiewicz (Poland) Some subclasses of K-valent Meromorphic functions. Convolution of subordination for some classes of regular functions. Dr. V. Karunakaran (India) Periodic Generalised functions Dr. G.P. Kapoor (India) Extreme points and support points of families of analytic functions. Prof. M. ObradovU (Yugoslavia) Some criteria for univalency in the unit disc. Dr. G.L. Reddy Linear Methods in Geometric Function Theory.
Session3
Chairman :Prof. H. Saitoh (Japan)
14.00- 15.30
Prof. R.M. Goel (India) Differential inequalities and local valency. Prof. O.P. Junl!ja (India) Univalent Functions with univalent GelfondLeontiev derivatives. Dr. Sampath Kumar (India) A survey of some recent results in univalent harmonic functions.
xviii
Session 4 Chairman 15.45 - 17.25 Dr. SL. Shukla Dr. KK. Dixit Dr. T. Ram Reddy Dr. M.S. Ganesan Dr. S.R. Kulkarni Ms.S.Latha Mr. Uday Naik Mr. S.B. Joshi Mr. K Satyanarayana
:Prof. R. W. Barnard (U.S.A) p-valent Bazilevit integral operators An application of fractional calculus. Neighbourhoods of analytic functions. Coefficient regions for polynomials starlike of order a. Radius ofconvexity and starlike ness for certain classes of analytic functions with (u:ed second coefficient. A generalized class of analytic functions. Generalized prestarlike functions. Starlike meromorphic functions with positive coefficients. A note on the Quasi Hadamard product of certain classes of starlike functions.
July 29, 1990
Session 1
Chairman : Prof. R.M. Goel (India)
9.00-10.30
Prof. H. Silverman (U.S.A) Open problems on univalent function with negative coefficients. Prof. T. Bulboacil (Romania) An extension of some classes of univalent functions.
Session 2
Chairman :Prof. Sampath Kumar (India)
10.45- 12.15
Prof. H. Saitoh (Japan) Some properties of certain multivalent functions. Prof. G. Dimkov (Bulgaira) Some properties of polynomials with negative coefficients. Dr. A.K Mishra (India) Convex hulls and extreme points of families of mulitvalent functions. Prof. T .K. Puttaswamy (U.S.A) A generalization of a Theorem ofW.B. Ford. Dr. Ponnusamy (India) First order differential inequalities in the complex plane.
Session3
Chairman :Prof. V. Singh (India)
13.00 - 14.00
Discussion "Open Problems" Closing Ceremony
xix
Illsf of Pdl'flcipants I Prof. Haji Mohammed, Kabul University, Kabul, Mghanistan. Prof. Shamsun Nahar, Eden University College, Bangaladesh. Prof. G.M. Dimkov, Bulgarian Academy of Sciences, 1090, Sofia, Bulgaria. Prof. M.A. Nasr, Mansoura University, Mansoura, Egypt. Prof. R. Niikki, University of Jyvaeskylae, Finland. Prof. S. Owa, Kinki University, Higashi-Osaka, Osaka- 577, Japan. Prof. H. Saitoh, Gunma College ofTechnology, 580, Tosiba- Machi, Maebashi, Gunma 371, Japan. Prof. Rosh ian Md. Ali, University Sains Malaysia, Mindem, Pulan, Pinang, Malaysia. Prof. Bruce O'Neil, ITM/Indiana University, Sek. 17, Shah Alam, Malaysia. Prof. K.M. Swe, Yangon University, Yangon, Mayanmar (Burma). Prof. Asif Ali Kazi, Institute of Mathematics and Computer Sciences, Sind University, Janshora, Sind, Pakistan. Prof. J. Stankiewicz, Technical University ofRzeszow, 35959, RZESZOW, P.O. Box 85, Poland. Dr. T. Bulboaca, Cluj-Napoca University, 3400 Cluj- Napoca, Romania. Dr. I.V. Anandam, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia. Prof. J.M. Anderson, University College, London WC 1, U.K Prof. D.K. Thomas, University College ofSwansea, Singleton Park, Swansea, SA28 PP, U.K Prof. R.W. Barnard, Texas Tech. University, Lubbock TX 79409, U.S.A. Prof. E.P. Merkes, University of Cincinnati, Cincinnati OH 45221, U.S.A. Prof. C.D. Minda, University of Cincinnati, Cincinnati OH 45221, U.S.A. Prof. S.S. Miller, State University of New York, College at Brockport, Brockport, New York 14420, U.S.A. Prof. T.K. Puttaswamy, Bull State University, College of Science and Technology, Muncie, Indiana 4 7306 - 0490, U.S.A. Prof. H. Silverman, College of Charleston, Charleston, SC 29424, U.S.A. Prof. E.M. Silvia, University of California, Davis, CA 95616, U.S.A. Prof. St. Ruscheweyh, Mathematics Institut Der Universitat, Am Hubland, Universitat Wiirzburg, B- 8700 Wiirzburg, West Germany. Prof. M. Obradovi~, Faculty ofTechnology and Metallurgy, 4, Karnegieva Street, 11000 Belgrade, Yugoslavia.
XX
-------;
~f. P. Achuthan, Indian Institute of Technology, Madras 600 036, India. Ms. Anbuchelvi,154/4, I.C.F. West Colony, Madras- 600 038, India. Ms. G. Ashok Kumar, No. 1-1-571115, Golconda X Road, Hyderabad - A.P. 500 020, India.
Prof. P. Balasubramaniam, Madras Christian College, Tambaram, Madras- 600 059, India. Dr. R. Bharati, Loyola College, Madras - 600 034, India. Dr. S.S. Bhoosnurmath, Kamatak University, Dharwad- 580 003, India. Ms. S. Chandra, Ramanujan Institute, University of Madras, Madras- 600 005.
Mr. Devadas Manjunath, B.V.B. Engineering College, Hubli, Kamatak State.
Prof. K.K. Dixit, Janta College, Bakewar, (Etawah) 206 124, U.P. India. Mr. Ferozekhan, Mazhrul Uloom College, Ambur (N.A. Distritct), India.
Dr. M.S. Ganesan, A.V.V.M. Sri Pushpam College, Poondi, Thanjavur Dt., India. Mr. A. George, 38, Second Street, Pudupet, Madras - 600 002, India.
Dr. R.M. Goel, Punjabi University, Patiala, 147 002, India. Mr. B. Gokuldas Shenoy, Mangalore University, Mangalagangotri- 574 199, India. Dr. S.K. Gupta, Member Secretary NCMER, Director, Department of Science and Technology, New Delhi, India. Ms. S. Indira, 16, Bhaskara Street, Rangarajapuram, Madras- 600 024, India. Ms. Jayamani, Arulmigu Palani Andavar College, Palani, India.
Mr. L. Jagannathan, A.V.V.M. Sri Pushapam College, Poondi, Thanjavur District, India. Mr. A.K. Joshi, Kamatak University, Dharwad- 580 003, India. Mr. S.B. Joshi, Walchand College ofEngineering, Sangli- 416 415, Maharashtra, India.
Prof. O.P. Juneja, Indian Institute of Technology, Kanpur, 280 016, India. Dr. G.P. Kapoor, Indian Institute ofTechnology, Kanpur- 206 016, India. Dr. V. Karunakaran, Madurai Kamaraj Univeristy, Palkalai Nagar, Madurai- 625 021, India. Dr. M.S. Kasi, Loyala College, Madras - 600 034, India. Ms. Kausalya Chari, Sri Subramania Swamy Govt. Arts College, Tirutani, 631 209, India.
Dr. S.R. Kulkarni, Willingdon College, Sangli- 416 415, Maharashtra, India. Ms. V.S. Kulkarni, Walchand College of Engineering, Sangli- 416 415, Maharashtra, India.
xxi
Ms. N. Lalitha, Syodji Street, Tripliane, Madras - 600 005, India. Ms. S. Latha, Maharaja's College, Mysore, Kamataka, India. Dr. S.Leeladevi,30, Telugu ChettyStreet, Woriur(P.O), Tiruchirapalli- 620003, India.
Mr. Lakshmi Narayanan, Madras Christian College, Tambaram, Madras - 600 059, India. Prof. P.V. Madhavan Nair, Government College, Kasaragod, Kerala, India. Dr. A.K. Mishra, Berhampur University, Bhanja Bahar- 760 007, Orissa, India.
Mr. M.S. Murthy, Madras Christian College, Tambaram, Madras - 600 059, India. Prof. S. Nag, Institute of Mathematical Sciences, Madras- 600113, India. Ms. Nalinakshi Madhavamurthi, Stella Maris College, Madras - 600 086, India.
Prof. K.S. Padmanabhan, M 7312, 31st Cross Street, Besant Nagar, Madras- 600 090, India. Dr. R. Parvatham, Ramanujan Institute, University of Madras, Madras - 600 005, India. Dr. S. Ponnusamy, SPIC Science Foundation, 92 G.N. Chetty Road, T.Nagar, Madras - 600 017, India.
Ms. Parvati Sahu, Berhampur University, Bhanja Bahar- 760 007, Orissa, India. Prof. Prem Singh, Indian Institute of Technology, Kanpur- 208 016, India. Dr. Premalatha Kumaresan, Ramanujan Institute, University of Madras, Madras600 005, India. Dr. S. Radha, J.B.AS. Women's College, Madras- 600 018, India.
Mr. Radha Krishnan Nair, Cusat, Cochin - 22, India. Ms. Rajalakshmi Rajagopal, Loyola College, Madras - 600 034, India.
Mr. Ramachandran, Vivekananda College, Madras - 600 004, India. Mr. R. Ramakarthikeyan, Vivekananda College, Madras - 600 004, India. Dr. T. Ram Reddy, Kakatiya University, Warangal - 506 009, A.P., India.
Prof. M.S. Rangachari, Ramanujan Institute, University ofMadras, Madras- 600 005, India. Prof. G. Rangan, Ramanujan Institute, University ofMadras, Madras- 600 005, India. Dr. K.N. Ranganathan, Vivekananda College, Madras- 600 004, India. Dr. G.L. Reddy, University ofHyderadabad, Hyderabad- 500134, India.
Ms. Rekha, Indian Institute of Technology, Madras- 600 036, India. Ms. Thomas Rosy, Madras Christian College, Tambaram, Madras- 600 059, India.
xxii
Dr. Sampath Kumar, Mangalore University, Mangalagangothri, Mangalore- 574199, India. Mr. S. Sampath Kumar, Ramanujan Institute, University of Madras, Madras- 600 005, India. Mr. S.K. Saraswat, Janta College, Bakewar District, Etawah - 206 124, U.P., India. Mr. K. Sathya Narayana, Kakatiya University, Vidyarangapuri, Warangal- 506 009, A.P., India.
Dr. S. L. Shukla, Janta College, Bakewar, Etawah - 206 124, India. Prof. V. Singh, Punjabi University, Patiala- 147 002, India.
Dr. D.K. Sinha, University of Calcutta, 92, AP.C. Road, Calcutta- 700 009, India. Dr. N.S. Sohi, G.N.D. University, Amritsar- 143 005, Punjab, India. Mr. C. Somanatha, Karnatak University, Dharward- 580 003, India. Mr. V. Srinivas, Indian Institute ofTechnology, Kanpur- 208 016, India. Mr. S. Srinivasan, Presidency College, Madras - 600 005, India. Mr. T.V. Sudarsan, S.I.V.E.T. College, India. Prof. K.G. Subramanian, Madras Christian College, Tambaram, Madras - 600 059, India.
Mr. S. Nayak, Indian Institute of Technology, Kanpur- 208 016, India. Ms. Suman Bhasin, Christ Church College, Kanpur- 208 001, India.
Mr. S.R. Swamy, J.M. Institute of Technology, Chitradurga, Karnataka, India. Prof. J. Thangamani, Stella Maris College, Madras - 600 086, India.
Mr. V. Thirugnana Sambandam, Govt. Arts College, Nandanam, Madras- 600 035, India. Mr. U.H. Naik, Willingdon College, Sangli 416 415, Maharashtra, India.
Dr. B.A. Uralegaddi, Karnatak University, Dharwad- 580 003, India. Ms. Vasuki, Ramanujan Institute, University of Madras, Madras- 600 005, India.
Mr. Veerapandian, A V.V.M. Sri Pushpam College, Poondi, Tanjore District, India. Mr. B.M. Verma, Janta College, Bakewar, Etawah District, U.P. - 206 124, India.
Dr. A. Vijayakumar, Cochin University, Cochin- 682 022, India. Dr. Vinod Kumar, Chirst Church College, Kanpur- 208 001, India. Prof. C. Viola Devapakkiam, Presidency College, Madras - 600 005, India.
xxiii
I
[CONTENTS
I
I V.Anandam
Functions of Generalized Bounded Argument Rotation
1
R. Bharathi and Rajalakshmi Rajagopal Meromorphic Functions and Differential Subordination
10
T.BulboacA An Extension of Certain Classes of Univalent Functions
18
G.Dimkov On Properties of Polynomials of Special Types
26
R.M. Goel Differential Inequalities and Local Valency
29
M. Jahangiri, H. Silverman and E.M. Silvia Classes of Functions defined by Subordination
34
R.A. Kortram A Note on Univalent Functions with Negative Coefficients
42
W. Ma, D. Minda and D. Mejia Distortion theorems for hyperbolically and spherically k-convex functions
46
E.P. Merkes On Integral Transforms of Jakubowski Functions
55
S.Nag Diff (Circle) and the Teich muller Spaces :A Connection Via String Theory
57
Nanjunda Rao and S. Latha A Generalized Class of Analytic Functions
M.A. Nasr and H.R. El. Gawad On the Fekete-Szegii problem for close-to-convex functions of order p
60
66
R. Parvatham and S. Srinivasan Study of Ruscheweyh Integral Operators on Certain Classes ofMeromorphic Functions
75
S. Ponnusamy First order Differential Inequalties in the Complex Plane
XXV
81
T.K. Puttaswamy A Generalization of a Theorem ofW.B. Ford
90
St. Ruscheweyh and L.C. Salinas On a Boundary Value Problem for Convex Univalent Functions II
96
H. Saitoh Some Properties of Certain Multivalent Functions
103
Sampath Kumar A Survey of some recent results in the Theory of Harmonic Univalent Functions
109
H. Silverman Conjectures for Univalent Functions with Negative Coefficients
116
V.Singh Some Inequalities for Starlike and Spiral-like Functions
125
V. Srinivas, G.P. Kapoor and O.P. Juneja Extreme Points and Support Points of Certain Class of Analytic Functions
136
J. Stankiewicz and Z. Stankiewicz Some Subclass of k-valent Meromorphic Functions
140
D.K.Thomas Bazilevi~
Functions with Logarithmic Growth
146
B. A. Uralegaddi Certain Subclasses of Analytic Functions
159
Open Problems
162
i
I
L------~-----~ ----------xxvi
-
------
-----------'
Functions of Generalized Bounded Argument Rotation
LV. Anandam Faculty of Sciences King Saud University P.O. Box 2455, Riyadh 11451, SAUDI ARABIA
L Introduction N ailah A. Al-Dihan has obtained many results concerning functions of bounded boundary rotation. Choosing from among them, we have presented here some theorems with a direct bearing on the interesting paper of Padmanabhan and Parvatham dealing with a class of functions of bounded boundary rotation. Not all theorems have complete proofs; but this should not come in the way of understanding how the functions of bounded argument rotation are generalized and their properties obtained.
2. Preliminaries Let E be the unit disc. An analytic function p (z) defined on E and of the form p (z) = 1 + 2.anzn is said to be in the class P [A, B],- 1 S B (k/2) (A- B).
i.e.,
Note: That this result is sharp can be seen by considering
f
(z)
= {
z(1 + B 1)1z)(A -B) (k + 2)1(4B) , (1 _ Bl) 2z) (A -B ) (k + 2)/(48 )
I lit I
UAA~~
B=~
= I~ I = 1,
B
*0
Proposition 4. Let f e R,. [A, B]. Then
f (z) I < {[k (A -B) sin-1 Br]/(2B), B * o, Iarg -z(kAr]l2r, B = 0.
Proof: This result can be proved by using the corresponding result for S* [A, B].
2
Theorem 5: Let fJ, ... , fn E Rk {A, B); AJ, ... • An real positive such that IA; = 1; a 1, complex such that I a; I 5 1 'v' i; and c is an arbitary constant. Then F (z) = c (fi (alz))A1 ... 0. Then
r.!.
••2
A-8
r
f(1-Bt) CiiJ·
lao
(1
+ Bt)
A
-4-
t
~-1
dt
~8 • • ~ 2
[J
'-MI
1
-
-
a
e2a
I --1
ta
l
J
~
r
A-8
I .!.J(1 + Bt) Cijj•
l
a
(1 - Bt)
A
]a
aJe
[ 1
0
These inequalities are sharp. 5
r Ml
I (l-1
~
t
-4-
~8 • • ~ 2
~lf(z)l ~
dt
••2
l
t
~-1
a
dt
, B = 0.
dt
la
J,
B "' 0,
Proof. By Theorem 10, there exists g e Rlt [A, Bl such that
,
~,
r"· j(F "· "'r
=
Take z = rei9 and integrate along the line t = pei9, 0 S p S r, to obtain
f(gr-
1
lf(z)l1/a =
-
a
f( r
(p il/\)1/a _e -, dp
P
0
1
S-
a
1 + Bp) (A- B) (It+ 2)/(4aB) =-.:....::L-"--;-----,--,,-------,-,,----,-plla-1 dp, (1 -Bp)(A-B)(k+2)1(4aB)
0 (g
=
(k + 2)/4 ::(It _ 2)/4 , s1o s2 e
)
S* [A, Bl
, (using the properties of S* [A, B], See Preliminaries).
This leads to the right-side inequality forB t:. 0 and similarly forB = 0. For the left-side inequalities: Let d be the radius of the open disc contained in the image of E by (j (z)) 11a. Let z 0 be a
I
I
point of I z I = r < 1 for which (j (z)) 11a value increases with r and is less than d.
assumes its minimum value. This minimum
Hence the line segment l connecting the origin with the point f lla (zo) will be covered entirely by the values off lla (z) in the image domain. Let y be the arc in E which is mapped by ro = (j (z))11a onto this line segment. Then for Bt:.O,andforanyz, lzl =r, lf(z)l 11a = lf(zo)l 11a = f ldrol = f ld (j(z))l 11a y
p
z
But (j (z))1/a =
!. a
f(g (t))1/a dt t 0
Hence from (1) and (2) l/(z)l1/a
~
(!_lg(z)llla ldzl > !. ) ix lz I - a
f( r
1 _ Bp) (A -B) (It+ 2)/(4aB)
( 1 + Bp)(A- B) (It+ 2)/(4aB)
t 1/a-1 dp
0
which leads to the left-side inequality forB t:. 0 and similarly forB = 0. a
For the sharpness of the inequalities, consider f e MIt [A, Bl .
f( r
1 + Bt) (A- B) (It+ 2)/(4aB) .:...._-~~---::-:---:-::-- p lla- 1 dt, B t:. 0,
-1 a
F(z) =
( 1 _ BdA- B) (It + 2)/(4aB)
0
r
~
J
e kAt12a
t 1/a-1 dt,
0
6
B
=0.
Theorem 12: (a- convexity). The radius of a- convexity for IE M: fA. B) is r (1, 0) as in Theorem 6. Proofi Now (1- a)
z['C$)
I
($['($))'
(z) +a j'C$)
=P ($) e P, [A,B] and Rep($)> Ofor lzl < r (1, 0) as
in the proof of Theorem 6. Hence the theorem.
5. Functions of generalized bounded argument rotation Definition 13: A normalized function I (z) = z + aaz2 + ... defined on E is said to be a function of generalized bounded argument rotation, noted I E T, [A, B, C, D), -1 5 B 0. We say that 8 = (gl' 8 2, ••• , 8nl belongs to the class I. (n; a; h).
Remark. Putting n = 1, the class I. (n; a; h) reduces to the class M 8(J (h) introduced by Padmanabhan and Manjini [4]. Theorem 1. Let 8 = (g 1 , 8 2 , ... ,gn} e I. (n; a; h) and let G be the arithmetic mean of 8 1 ,g2 , ... ,gn. Then G satisfies the condition (Ka
-z (Ka
Proof: Since 8 = 181'82, ...
* G)' (z) * G) (z)
-< h (;:),
z e E.
(1)
e I. (n; a; h), for any zoe E, (Ka • g;Y (zo) -zo n e h (E),
,gn}
.!.
L,' (zo ) =::.c1:..__ _ _ _ _ _
.:._i
1
=
n
n
l: h
((•Ji).
i =1
- :E (Ka •g) (zo) n i=1
Now,
we have
i~1
O), i=1,2, ... ,n.
0
11
If his bounded in E andRe (y + 2) >max Re h (z), then G = (Gl' G2 ,
..•
,G,.)e I. (n; a; h).
zeE
Proof: From the definition of G; (z) it follows that zG'; (z) + (y + 2) G; (z) = (y + 1) 8; (z), and on taking convolution with Ka we obtain z (Ka
* G;)' (z) + (y + 2) (Ka * G;) (z) = (y + 1)
p (z )
-1 L" P; + (a + 1)
n
h( )
- - - - ' " ' - - - - - - _ _L..!L =~
n
n i=1
for some Cl)j e E . . Since h is convex, there exists a - -1 z 0
" p: (z ) 2. 0
n i=1 ' _ _---..!..:::..!....__ _ 1
+-
L" P; (z) + (a
n i=1
(J)o
,.
1 --
2.
n ;- 1
+ 1)
-
~-
=
e E, such that
" h (ro;) 2. i=1 .:..=._ _
n
'
=h .
(J)o E
E.
1 " Setting Q (z) = - L P; (z), we have n i=1
(=~;~~ + 1) + Q(z)
-Q
-< h (z),
which by Lemma 1, impliesthat Q (z)-< h (z). From (6)
-Q
(z) +(a+ 1)
-P; (z) -< h (z),
where Q (z) -< h (z). An application of Lemma 2 gives -pi (z) -< h (z), which implies that g = lg1 , g 2, ... ,g,.l e I. (n; a; h).
3. The Class C* (n; a, h) Definition 2. Let C* (n; a, h) denote the class of all functions f e M such that
n
[-z (Ka *f)' (z)] " -
0.
0
Then by Theorem 2, G
= (G1 , G 2 , ... ,Gn) e l: (n; a, h). Let z(Ka *F)' (z) p(z) = -n- - = - - - -
I
*G) (z)
(Ka
)=1
Now, from the definitions of Gi and F z (Ka • Gi) (z) + (c + 2) (K0
•
Gi) (z) = (c + 1) (K0
- L (Ka + 1 * gi) (z)
-
ni=1
L (Ka * gi) (z)
ni=1
n
where g = (gl'g2 ,
... ,gn] e
I (n; a, h) and z
.L Proof: For a = 0, the theorem is trivial and hence we can assume that a (z) =
P
,t.
0. Let
z (Ka *f)' (z)
1
n
n
i=1
- L (Ka *g) (z)
Then an easy calculation shows that + (z)= z (Ka +1 • f)' (z) p 1 n
zp'(z) 1
n
- L Qi (z) + (a
where qi (z) =
z
nj=1 (Ka *g.)' n
1
L (Ka + 1 • gi) (z)
-
ni=1
1 n
(z)
'
- L (Ka * gi) (z) n
+ 1)
·Also-- L qi (z) ~ h (z). Hence n i•1
i =1
-az p' (z) S (a; f; gl' g2, .. ·• 6n) = - 1--n..::.:.::;...c:..._-'=-''----- p (z) ~ h (z), -
n
_Lqi(z)+(a+1) i=1
since f e I 0 (n; a, h). Now an application of Lemma 2 gives -p (z) ~ h (z) there by completing the proof. Theorem 7. For a> ~ ~ 0, rex (n; a, h) c
Proof: The case
~
Ill (n; a, h).
= 0 was treated in the previous theorem. Hence we assume that
f e rex (n; a, h) implies that
S (a; f; 6p g 2, ... , gn)
~
h (z), z e E.
~ ,t.
0,
(7)
By Theorem 6, we have -z (Ka *f)' (z) n ~ - L (Ka •gi) (z) n i=1
1
16
h (z).
(8)
From (7) and (8) it follows that -z 1 (Ka * f)' (z1 ) e h (E) n (Ka * gi) (z1 )
1
- 2. n
i=l
and
z
(K 0
- (1 - a) 1 \
*f)' (z ) 1
e h (E).
-n i=l 2. (Ka * gi) (zl)
Now h is convex and 1i < 1 and hence we have a S (p; f;gl'g2, ... ,gn) (zl)
E
h (E),
there by showing f e l:~ (n; a, h).
References 1. P. Eeingenberg, S.S. Miller, P.T. Mocanu and M.O. Reade, On a Briot-Boquet Differential subordination, General inequalities, 3 (Birkhauser Verlag-Basel), 339-348. 2. K.S. Padmanabhan and R. Parvatham, Some applications of differential subordination, Bull. Austral. Math. Soc. 37 (1985), 321 -330. 3. K.S. Padmanabhan and R. Parvatham, Analytic functions and differential subordination, Complex Analysis and applications 85, Sofia, 1986. 4. K.S. Padmanabhan and R. Manjini, Differential subordination and Meromorphic functions, Annales Polon. Math. 49 (1988), 21 - 30.
17
An Extension of certain classes of Univalent Functions
T. Bulboaca Department of Mathematics, University of Cluj-Napoca, 3400 Cluj-Napoca, ROMANIA.
1. Introduction Let A be the family of functions f which are analytic in the unit disc U = (z e C: I z I < 1) and normalized with f (0) = 0, f' (0) = 1. In [1 0] S. Ruscheweyh defined the classes Kn of functions f e A which satisfy Re ( D" + 1 f (z)ID" f (z)J > 1/2, z e U. where nn f (z) = z/(1 -z)R + 1 * f (z) = z a, z E u, where 0 Sa< 1. He proved that Rn + 1 (a) c Rn (a), n e No= (0, 1, 2, ... I and solved the problem of the radius of Rn + 1 (a) in Rn (a); note that the classes Rn = Rn (0) were studied by R. Singh and S. Singh [12]. Related classes obtained by convex combinations using the Ruscheweyh's differential operator D"f were studied by the author in [2], [3), [4), [5]. In this paper we introduce some other new classes of analytic functions obtained by the above method which generalize the classes Rn (a); we will extend some inclusion results by using sharp subordination results and several particular cases will also be given.
2. Preliminaries We denote by KRn, Re {(1 - a)
tt
(5) the class of functions f e A which satisfy
z(Dnf(z))' nn+2f(z)} nn f (z) + ann + 1 f (z) > 5,
z
E
u where a~ 0, 5 < 1.
Note that KRn.o (5) = Rn (5), KRn,1 (5) = Kn+1 (5) where Kn+1
=
=
(~) = Kn+1
[10],
KRo,o (5) S* (5) and KRo, 1 (5) sc (25- 1) where S* (5) and sc (25- 1) represent the classes of starlike and convex functions of order 5 and 25 - 1 respectively. We easily deduce, using the relation 18
z (U' f ~))' = (n + 1) U' + 1 f ~) - n U' f ~>.
(1)
thatRn (!)) =Kn (::
~)
or Kn (!)) =Rn
~e
U, ne No>.
((n + 1)1)-n), henceRn r- ; n) =K;. ~) eKn.
Let ARn (I)) be the class of functions f e A which satisfy lY' + 1 f (z) Re nn + 1 g ~) > li, z E
u whereg E Rn (I)), I)< 1.
We have thatRn (!))eARn (li), n e No. Let f and g be regular in U; we say that f is subordinate tog, written f is univalent in U, f (0) = g (0) and f (U) !:;;; g (U).
~)
-< g
~),
if g
We shall use the following theorems to prove our results. Theorem A. [6, Theorem 1]. Let {3, r e C, let h (z) = c + h 1z + ... be a convex (univalent) function in u. with Re ({3 h (z) + r) > 0, z E u. If p (z) = c + p JZ + ... is analytic in U, then p
~>
+ p zp r~z) z +y -< h
~>
impliesp ~) -< h
~).
Theorem B. [8]. Let h, q be two analytic and univalent functions in U and suppose that q is analytic in
U.
If 'Y: C 3 -+ C satisfies:
(a) 'I' is analytic in a domain D c C3, (b) (q (0), 0, 0) E D and 'I' (q (0), 0 , 0) E h (U), (c) 'I' (r, s, t) E D when (r, s, t) e D, r = q
. s = m Cq'(C) and
Re (1 +tis) 2: m Re (1 + Cq" 0 in U. Q (-r, t) is real andRe (1/Q (z, t)) :?!1 /Q (-r, t) for lzl Sr< 1 and t e [0, 11. 1
1
1
lfQ ~) = J Q (z, t) dJ,t (t), then Re Q (z) 2: Q (-r) , for lz I ~ r. 0
Theorem D. [6]. Let p > 0, p + y > 0 and -y!p S I) < 1. Then the differential equation . l ent sol utJon . . U., gJven . by q '-) v. + A zq'(z) ( ) = 1-(1-21i)z m , q (0) = 1, h as a unwa 1 I'
q
Z +
y
+Z
1
__ 1_ :1. -J(1-zj(1-ll) P+y-1 1-(1-21i)z q ~) - PQ(z) - p , where Q ~) - 0 1 _ tz t dt, z e U and q ~) -< . 1 +z lfp
~)
= 1 + P1Z + ... is regular in U and satisfies the differential subordination
'-) z p' (z) 1 - (1 - 25) z p""+pp(z)+y-< 1+z then p (z) -< q (z) and this subordination is sharp.
19
3. Main Results Th
eorem
then
f
LH'O 1
n+2 Sa 0, hence applying Theorem D we conclude that (3) has the univalent solution
1
q (z) = p Q (z)- ~;where Q b > 0 from (3), by using (4), (5) and (6) we deduce
20
1
Q(z) = (1-z)" j(l-tz)- /(z) +PU+ /'(z)> >li-(1-li)p,zeU implies that Re zf'(z) > j(z)
1
F( 1 ,2(1-li~(l+P>,1PP;~,ze
U
and this result is sharp. Note that this implication represents Corollary 3 from [2] and taking a= PICP + 1) we obtain the order of starlikeness of P-convex functions for P > 1. Taking a= 1 in Theorem 1 and using Kn (li) = Rn ((n + 1) li-n) we obtain: Corollary 2. If -n/2 s; a < 1, n
EN o.
then f
E
Rn
+1
(li) impliPs f
E
* where Rn (lin),
.,* n+1 d h" l · ha un=F(1, 2 (1 -Ii),n+ 2 ;1!2 ) -nan t ISresutiSS rp. Remarks 1. This result represents an improvement and an extension of the Ahuja's result [1 l which proved that Rn + 1 (li) c Rn (li) for 0 s; li < 1. 2. Another version of this result is: if 1/2 s; li < 1 then Kn + 1 (li) c Rn (li • (1)), n
•
n+1
where lin (1) = F (1 , 2 (n + 2) (1 _ li), n + 2 ; 112) - n and this inclusion is sharp.
Theorem2.If-n 58< 1 thenRn
+ 1 (6)
eRn (6), n
E
No.
Proof: Since Rn (li) = Kn ((n + li)!(n + 1)) then
f Taking
E
Rn
+1
nn + 2 f (z) n + 1 +a (li) if and only if Re nn + 1 f (z) > n + 2 , z E
u. 0
nn + 1 f (z) nn f (z) - p (z) thenp (0) = 1 and using (1) we obtain that nn+2j(z) 1 ( zp'(z) ) nn + 1 f (z) = n + 2 (n + 1) p (z) + p (z) + 1
and denoting P(z) = (n + 1) p (z)- n -li, we deduce that f e Rn + 1 (li) is equivalent to zP' (z)
Re { p (z) + p (z) + n +a
}
> 0, z E U;
where h (z) "' ha (z) -li and ha (z) =
.
zP' (z)
I.e., p (z) + p (z) + n +a
1-(1-2li)z . 1 +z 22
-< h (z)
Because h (z) is convex (univalent) in U
andRe(~
y:n + S, by using Theorem A we conclude that P (z)
h (z) + y) > 0, z e U where
~
= 1 and
n+S -< h (z) or Re JY&+1J(z) D" I (z) > n +1 , z e U.
Hence f e R,. (S). Note that this result represents also an extension of Lemma 1 from [1 ].
Theorem 3. If -n s;; < 1 thenAR,. + 1 (6) cAR,. (6), n
E
No.
D" + 2 I Cz>
Proof. Iff e AR,. + 1 (S), there exists a g e R,. + 1 (S) such that Re D" + 2 g (z) > S, z e U and by Theorem 2 we have g e R,. (S). Let p (z) = D" + 1 f (z)ID" + 1 g (z), p (0) = 1; using (1) a D" + 2 f (z) simple computation shows that D" + 2 g (z) = p (z) + a (z) z p' (z ), where 1 D" + 1 g (z) D" + 2 g (z) n + 1 + S a (z) =n + 2 D" + 2 g (z). Because g e Rn+1 (S) we have Re D" + 1 g (z) > n + 2 > 0, z e U. hence Re a (z) > 0, z e U. Then f e AR,. + 1 (S) is equivalent to 1- (1- 2S)z p (z) + a (z) zp' (z) h (z) "' . 1 +z
-
Szmplzes Re g(z) > S,ze U whenever g
E
s• (S).
Corollary 4. If -1 s S < 1 and f, g e A, then
Re
/'(z) + ~zf"(z) 1
g'(z)+ 2zg"(z)
f'(z)
>SimpliesR eg'(z) > S,ze U
whenever g e sc (S). For S = 0 the result of Corollary 3 was obtained by K. Sakaguchi in [11 ]. n+2 n+
Theorem4.If0 Sa