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

MathSciNet review:
0387563

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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**0150310****[2]**F. Holland and D. K. Thomas,*On the order of a starlike function*, Trans. Amer. Math. Soc.**158**(1971), 189–201. MR**0277705**, 10.1090/S0002-9947-1971-0277705-5**[3]**R. R. London and D. K. Thomas,*An area theorem for starlike functions*, Proc. London Math. Soc. (3)**20**(1970), 734–748. MR**0262481****[4]**Ch. Pommerenke,*On starlike and convex functions*, J. London Math. Soc.**37**(1962), 209–224. MR**0137830****[5]**T. Sheil-Small,*Starlike univalent functions*, Proc. London Math. Soc. (3)**21**(1970), 577–613. MR**0276460**

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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