×

zbMATH — the first resource for mathematics

On certain analytic functions associated with Ruscheweyh derivatives and bounded Mocanu variation. (English) Zbl 1155.30007
The authors study the class \(\mathcal A\) of functions \(f\) having the form \(f (z) = z + \sum^{\infty}_{m=2} a_m z^m\) that are analytic in the unit disk \(E = \{z : |z| < 1\}\) and the subclasses of functions in \(\mathcal A\) that are univalent, close-to-convex, starlike, convex, of bounded radius rotation, of bounded Mocano variation, or Paatero bounded radius rotation.
The authors give a unified approach to the study via a huge new parametrized family of classes denoted by \(R_k (\alpha, a, \gamma)\), where \(k \geq 2,\alpha \geq 2, a > 0\), and \(0\leq \gamma < 1\). The definition of \(R_k (\alpha, a,\gamma )\) is complicated, but, for example:
\(R_k (0, 1, 0)\) is the class of bounded radius rotation;
\(R_2 (\alpha, 1, 0)\) is the class of alpha-starlike functions;
\(R_k (1, 1, 0)\) is the class of Paatero bounded boundary rotation;
\(R_k (\alpha, 1, 0)\) is the class of bounded Mocano variation;
\(R_2 (1, 1, 0)\) is the class of convex functions; and
\(R_2 (0, 1, 0)\) is the class of starlike functions.
The authors’ main results are inclusion relationships among the classes \(R_k (\alpha, a,\gamma )\) and a crieterion for the univalence of the Ruscheweyh derivatives of functions in some of these classes.

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bazilevic, I.E., On a class of integrability of the lowner – kufarev equation, Mat. sb., 37, 471-476, (1955)
[2] Coonce, H.B.; Ziegler, M.R., Functions with bounded mocanu variation, Rev. roumaine math. pures appl., 19, 1093-1104, (1974) · Zbl 0363.30010
[3] Goodman, A.W., Univalent functions, vols. I, II, (1983), Polygonal Publishing House Washington, NJ
[4] Miller, S.S.; Mocanu, P.T., Differential subordinations: theory and applications, Pure appl. math., vol. 225, (2000), Marcel Dekker New York · Zbl 0954.34003
[5] Inayat Noor, K., On subclasses of close-to-convex functions of higher order, Int. J. math. math. sci., 15, 279-290, (1992) · Zbl 0758.30010
[6] Inayat Noor, K., Properties of certain analytic functions, J. nat. geom., 7, 11-20, (1995) · Zbl 0911.30011
[7] Inayat Noor, K., On some subclasses of functions with bounded boundary and bounded radius rotation, Panamer. math. J., 6, 75-81, (1996) · Zbl 0966.30007
[8] Pinchuk, B., Functions with bounded boundary rotation, Israel J. math., 10, 7-16, (1971) · Zbl 0224.30024
[9] Ruscheweyh, S., Convolution in geometric function theory, (1982), Les Presse de Universite de Montreal Montreal
[10] Ruscheweyh, S.; Singh, V., On certain extremal problems for functions with positive real parts, Proc. amer. math. soc., 61, 329-334, (1976) · Zbl 0347.30007
[11] Shiel-Small, T., On bazilevic functions, Q. J. math., 23, 135-142, (1972) · Zbl 0236.30019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.