Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0575.30019
Miller, Sanford S.; Mocanu, Petru T.
On some classes of first-order differential subordinations. (English)
[J] Mich. Math. J. 32, 185-195 (1985). ISSN 0026-2285

The paper contains several results of the following type: Theorem. Let h be analytic in $U=\{z:$ $\vert z\vert <1\}$, let $\phi$ be analytic in a domain D containing h(U) and suppose a) Re $\phi$ (h(z))$>0$, $z\in U$ and either b) h(z) is convex or b) $H(z)=zh'(z)\phi (h(z))$ is starlike w.r.t. the origin. If p is analytic in U with $p(0)=h(0)$, p(U)$\subset D$, and $$ p(z)+zp'(z)\phi (p(z))\prec h(z), $$ then p(z)$\prec h(z)$. An application to univalent functions is given: Let f be analytic in U, $f(0)=0$ and either $$ \vert f''(z)/f'(z)\vert \le 2\quad or\quad \vert zf''(z)/f'(z)+1\vert <2,\quad (z\in U). $$ Then f is starlike in U.
[J.Waniurski]

MSC 2000:: *30C45 Special classes of univalent and multivalent functions
30C80 Maximum principle, etc. (one complex variable)

Keywords: differential subordination; convex; starlike

Cited in: Zbl 1162.30312 Zbl 1152.30316 Zbl 1069.30026 Zbl 0976.30014 Zbl 0853.30011 Zbl 0810.30016 Zbl 0748.30017 Zbl 0748.30016 Zbl 0685.30006 Zbl 0672.30022 Zbl 0728.30010 Zbl 0674.30004

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]