Radii problems for generalized sections of convex functions
HTML articles powered by AMS MathViewer
- by Richard Fournier and Herb Silverman PDF
- Proc. Amer. Math. Soc. 112 (1991), 101-107 Request permission
Abstract:
A classical theorem of Szëgo states that for functions $f(z) = z + \Sigma _{k = 2}^\infty {a_k}{z^k}$ convex in $|z| < 1$, the sequence of partial sums ${f_n}(z) = z + \Sigma _{k = 2}^n{a_k}{z^k}$ must be convex in $|z| < \frac {1}{4}$. For the more general family consisting of functions of the form $z + \Sigma _{k = 2}^\infty {a_{{n_k}}}{z^{{n_k}}}$, where $\left \{ {{n_k}} \right \}$ denotes an increasing (finite or infinite) sequence of integers $( \geq 2)$, we find the radius of convexity $( \approx 0.21)$ and the radius of starlikeness $( \approx 0.37)$. The extremal function in both cases is $z + {z^2}/(1 - {z^2}) = z + \Sigma _{k = 1}^\infty {z^{2k}}$ associated with the convex function $z/(1 - z) = z + \Sigma _{k = 2}^\infty {z^k}$.References
- J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. of Math. (2) 17 (1915), no. 1, 12–22. MR 1503516, DOI 10.2307/2007212
- Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
- Richard Fournier, On neighbourhoods of univalent starlike functions, Ann. Polon. Math. 47 (1986), no. 2, 189–202. MR 884935, DOI 10.4064/ap-47-2-189-202
- A. W. Goodman and I. J. Schoenberg, On a theorem of Szegő on univalent convex maps of the unit circle, J. Analyse Math. 44 (1984/85), 200–204. MR 801293, DOI 10.1007/BF02790196
- Stephan Ruscheweyh, On the radius of univalence of the partial sums of convex functions, Bull. London Math. Soc. 4 (1972), 367–369. MR 316697, DOI 10.1112/blms/4.3.367
- Stephan Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981), no. 4, 521–527. MR 601721, DOI 10.1090/S0002-9939-1981-0601721-6
- St. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119–135. MR 328051, DOI 10.1007/BF02566116
- Herb Silverman, Radii problems for sections of convex functions, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1191–1196. MR 942638, DOI 10.1090/S0002-9939-1988-0942638-3
- G. Szegö, Zur Theorie der schlichten Abbildungen, Math. Ann. 100 (1928), no. 1, 188–211 (German). MR 1512482, DOI 10.1007/BF01448843
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 112 (1991), 101-107
- MSC: Primary 30C45; Secondary 30C50
- DOI: https://doi.org/10.1090/S0002-9939-1991-1047000-3
- MathSciNet review: 1047000