On a conjecture of Danikas and Ruscheweyh
نویسندگان
چکیده
We construct a holomorphic function f in the unit disc, whose derivative belongs to the Hardy class H1 , and the image of the unit circle under z 7→ ∫ z 1 f ′(ζ) dζ ζ is a simple curve, but f is not univalent.
منابع مشابه
Remarks on a Multiplier Conjecture for Univalent Functions
In this paper we study some aspects of a conjecture on the convolution of univalent functions in the unit disk D , which was recently proposed by Grünberg, Ronning, and Ruscheweyh (Trans. Amer. Math. Soc. 322 (1990), 377-393) and is as follows: let 3 := {/ analytic in D: \f"(z)\ < Ref'(z), z £ D} and g, h £ S? (the class of normalized univalent functions in D) . Then Ke(f*g*h)(z)/z > 0 in D . W...
متن کاملDisproof of a Conjecture on Univalent Functions
We disprove the Gruenberg-RRnning-Ruscheweyh conjecture, namely that Re d g z (z) > 0; jzj < 1; holds for g 2 S, the set of normalized univalent functions in the unit disk D , and d analytic with jd 0 (z)j Re d(z) in D , d(0) = 1. Here stands for the Hadamard product.
متن کاملA note on Fouquet-Vanherpe’s question and Fulkerson conjecture
The excessive index of a bridgeless cubic graph $G$ is the least integer $k$, such that $G$ can be covered by $k$ perfect matchings. An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless cubic graph has excessive index at most five. Clearly, Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5, so Fouquet and Vanherpe as...
متن کامل$L^p$-Conjecture on Hypergroups
In this paper, we study $L^p$-conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions for a weighted Lebesgue space $L^p(K,w)$ to be a convolution Banach algebra, where $1<p<infty$, $K$ is a locally compact hypergroup and $w$ is a weight function on $K$. Among the other things, we also show that if $K$ is a locally compact hyper...
متن کاملOn the oriented perfect path double cover conjecture
An oriented perfect path double cover (OPPDC) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each arc of $G_s$ lies in exactly one of the paths and each vertex of $G$ appears just once as a beginning and just once as an end of a path. Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete Math. 276 (2004) 287-294) conjectured that ...
متن کامل