zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On certain analytic functions with bounded radius rotation. (English) Zbl 1222.30011
Summary: Certain classes $R_k(\mu,\alpha)$; $k\ge 2$, $\mu>-1$, $0\le\alpha<1$ of analytic functions are defined in the unit disc using convolution technique. It is shown that functions in $R_{k}(\mu ,\alpha )$ are of bounded radius rotation. It is proved that $R_{k}(\mu ,\alpha )$ and some other newly introduced related classes are invariant under the generalized Bernardi integral operator. The converse case as a radius problem is also considered. Theorems proved in this paper are best possible in some sense.

30C45Special classes of univalent and multivalent functions
Full Text: DOI
[1] Bernardi, S. D.: Convex and starlike univalent functions, Trans. amer. Math. soc. 135, 429-446 (1969) · Zbl 0172.09703 · doi:10.2307/1995025
[2] Libera, R. J.: Some classes of regular univalent functions, Proc. amer. Math. soc. 16, 755-758 (1965) · Zbl 0158.07702 · doi:10.2307/2033917
[3] Livingston, A. E.: On the radius of univalence of certain analytic functions, Proc. amer. Math. soc. 17, 352-357 (1966) · Zbl 0158.07701 · doi:10.2307/2035165
[4] Carlson, B. C.; Shaeffer, D. B.: Starlike and prestarlike hypergeometric functions, SIAM J. Math. anal. 15, 737-745 (1984) · Zbl 0567.30009
[5] Ruscheweyh, St.: New criteria for univalent functions, Proc. amer. Math. soc. 49, 109-115 (1975) · Zbl 0303.30006 · doi:10.2307/2039801
[6] Pinchuk, B.: Functions with bounded boundary rotation, Israel J. Math. 10, 7-16 (1971) · Zbl 0224.30024 · doi:10.1007/BF02771515
[7] Goodman, A. W.: Univalent functions, Univalent functions , II (1983) · Zbl 1041.30501
[8] Singh, R.; Singh, S.: Integrals of certain univalent functions, Proc. amer. Math. soc. 77, 336-340 (1979) · Zbl 0423.30007 · doi:10.2307/2042182
[9] Miller, S. S.: Differential inequalities and Carathéodory functions, Bull. amer. Math. soc. 81, 79-81 (1975) · Zbl 0302.30003 · doi:10.1090/S0002-9904-1975-13643-3
[10] Miller, S. S.; Mocanu, P. T.: Differential subordinations, (2000) · Zbl 0954.34003
[11] Ruscheweyh, St.; Singh, V.: On certain extremal problems for functions with positive real part, Proc. amer. Math. soc. 61, 329-334 (1976) · Zbl 0347.30007 · doi:10.2307/2041336
[12] Noor, K. Inayat: Some properties of certain classes of functions with bounded radius rotation, Honam math. J. 19, 97-105 (1997) · Zbl 0953.30004
[13] Bulboca, T.: On particular n-${\alpha}$-close-to-convex functions, Studia univ. Babeş-bolyai math. 36, 71-75 (1991)