On two extremal problems related to univalent functions. (English) Zbl 0818.30013

Fournier, Richard; Ruscheweyh, Stephan

Rocky Mt. J. Math. 24, No. 2, 529-538 (1994).

Let \(\Lambda: [0, 1]\to \mathbb{R}\) be integrable over \([0, 1]\) and positive in \((0,1)\) and \(S\) the class of functions univalent in the unit disk \(D\) and normalized as usual. The authors consider for \(f\in S\), \[ L_ \Lambda(f)= \inf\Biggl\{\int^ 1_ 0 \Lambda(t)\;(\text{Re}(f(tz)/tz- 1/(1+ t)^ 2) dt\mid z\in D\Biggr\} \] and \(L_ \Lambda(S)= \inf\{L_ \Lambda(f)\mid f\in S\}\) resp. \(L_ \Lambda(C)= \inf\{L_ \Lambda(f)\mid f\in C\}\), where \(C\subset S\) denotes the subclass of closed-to-convex functions.
They ask whether there are functions \(\Lambda\) such that \(L_ \Lambda(S)= 0\) and show that for
\(\Lambda(t)/(1- t^ 2)\) decreasing on \((0,1)\), \(L_ \Lambda(C)= 0\). Furthermore they consider the class \(P_ \beta\) of functions \(f\) holomorphic in \(D\) normalized in the origin as usual for which \(f'(D)- \beta\) lies in a halfplane bounded by a straight line through the origin and functions \[ \lambda: [0, 1]\to \mathbb{R},\quad \int^ 1_ 0 \lambda(t) dt= 1,\quad \lambda\geq 0. \] They determine numbers \(\beta= \beta(\lambda)\) such that the conclusion \[ f\in P_ \beta\Rightarrow V_ \lambda(f) (z)= \int^ 1_ 0 \lambda(t) f(tz)/t dt\in S \] holds and for some special \(\lambda\) they find \(\beta= \beta(\lambda)\) for which \(V_ \lambda(P_ \beta)\subset S^*\), where \(S^*\) is the class of starlike functions. For \(\lambda(t)= (c+ 1) t^ c\), \(c> -1\), this solves a problem discussed before by many authors.

Reviewer: K.J.Wirths (Braunschweig)

MSC:

30C55

General theory of univalent and multivalent functions of one complex variable

References:

[1]	R. Ali, On a subclass of starlike functions , · Zbl 0816.30010
[2]	D.M. Campbell and V. Singh, Valence properties of the solution of a differential equation , Pacific J. Math. 84 (1979), 29-33. · Zbl 0387.34012
[3]	P.L. Duren, Univalent functions , Springer-Verlag, 1983.
[4]	H. Grunsky, Neue Abschtzungen zur konformen Abbildung ein- und mehrfach zusammenhngender Bereiche , Schr. Math. Inst. u. Inst. Angew. Math. Univ. Berlin, 1 (1932), 95-140. · Zbl 0005.36204
[5]	P.T. Mocanu, On starlikeness of Libera transform , Mathematica (Cluj) 51 (1986), 153-155. · Zbl 0635.30013
[6]	M. Nunokawa, On starlikeness of Libera transformation , Complex Variables Theory Appl. 17 (1991), 79-83. · Zbl 0758.30011
[7]	M. Nunokawa and D.K. Thomas, On the Bernardi integral operator , · Zbl 0991.30501
[8]	St. Ruscheweyh, Convolutions in geometric function theory , Les Presses de l’Université de Montréal, Montréal, 1982.
[9]	R. Singh and S. Singh, Starlikeness and convexity of certain integrals , Ann. Univ. Mariae Curie-Skłodowska 35 (1981), 45-47. · Zbl 0559.30007
[10]	——–, Starlikeness of close-to-convex functions , Indian J. Pure Appl. Math. 13 (1982), 190-194. · Zbl 0489.30009
[11]	——–, Convolution properties of a class of starlike functions , Proc. Amer. Math. Soc. 106 (1989), 145-152. · Zbl 0672.30007