On the partial sums of convex functions of order $1/2$
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- by Ram Singh PDF
- Proc. Amer. Math. Soc. 102 (1988), 541-545 Request permission
Abstract:
Let $f\left ( z \right ) = z + {a_2}{z^2} + \ldots$ be regular and univalently convex of order $1/2$ in the unit disc $U$ and let ${s_n}\left ( {z,f} \right )$ denote its $n$th partial sum. In the present note we determine the radius of convexity of ${s_n}\left ( {z,f} \right )$, depending on $n$, and generalize and sharpen a result of Ruscheweyh concerning the partial sums of convex functions. We also prove that for every $n \geq 1,{\text {Re}}\left ( {{s_n}\left ( {z,f} \right )/z} \right ) > 1/2$ in $U$.References
- Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494 A. Kobori, Zwei Satze über die abschnitte schlichter Potenzreihen, Mem. Coll. Kyoto 17 (1934), 172-186.
- M. S. Robertson, Univalent functions starlike with respect to a boundary point, J. Math. Anal. Appl. 81 (1981), no. 2, 327–345. MR 622822, DOI 10.1016/0022-247X(81)90067-6
- W. Rogosinski and G. Szegö, Über die Abschnitte von Potenzreihen, die in einem Kreise beschränkt bleiben, Math. Z. 28 (1928), no. 1, 73–94 (German). MR 1544940, DOI 10.1007/BF01181146
- 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
- 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
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 541-545
- MSC: Primary 30C45; Secondary 30C55
- DOI: https://doi.org/10.1090/S0002-9939-1988-0928976-9
- MathSciNet review: 928976