An asymptotic formula for an integral in starlike function theory

Authors:
R. R. London and D. K. Thomas

Journal:
Trans. Amer. Math. Soc. **215** (1976), 393-406

MSC:
Primary 30A32

DOI:
https://doi.org/10.1090/S0002-9947-1976-0387563-0

MathSciNet review:
0387563

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The paper is concerned with the integral

*f*is a function regular and starlike in the unit disc, , and the parameters are real. A study of

*H*is of interest since various well-known integrals in the theory, such as the length of , the area of , and the integral means of

*f*, are essentially obtained from it by suitably choosing the parameters. An asymptotic formula, valid as , is obtained for

*H*when

*f*is a starlike function of positive order , and the parameters satisfy . Several easy applications of this result are made; some to obtaining old results, two others in proving conjectures of Holland and Thomas.

**[1]**W. K. Hayman,*On functions with positive real part*, J. London Math. Soc.**36**(1961), 35-48. MR**27**#311. MR**0150310 (27:311)****[2]**F. Holland and D. K. Thomas,*On the order of a starlike function*, Trans. Amer. Math. Soc.**158**(1971), 189-201. MR**43**#3438. MR**0277705 (43:3438)****[3]**R. R. London and D. K. Thomas,*An area theorem for starlike functions*, Proc. London Math. Soc. (3)**20**(1970), 734-748. MR**41**#7087. MR**0262481 (41:7087)****[4]**Ch. Pommerenke,*On starlike and convex functions*, J. London Math. Soc.**37**(1962), 209-224. MR**25**#1279. MR**0137830 (25:1279)****[5]**T. Sheil-Small,*Starlike univalent functions*, Proc. London, Math. Soc. (3)**21**(1970), 577-613. MR**43**#2207. MR**0276460 (43:2207)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30A32

Retrieve articles in all journals with MSC: 30A32

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0387563-0

Keywords:
Starlike function,
order,
length,
area,
integral means

Article copyright:
© Copyright 1976
American Mathematical Society